2021
2021
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Paper 1, Section I, F
2021 commentState and prove the alternating series test. Hence show that the series converges. Show also that
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Paper 1, Section I, F
2021 commentState and prove the Bolzano-Weierstrass theorem.
Consider a bounded sequence . Prove that if every convergent subsequence of converges to the same limit then converges to .
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Paper 1, Section II, F
2021 comment(a) State the intermediate value theorem. Show that if is a continuous bijection and then either or . Deduce that is either strictly increasing or strictly decreasing.
(b) Let and be functions. Which of the following statements are true, and which can be false? Give a proof or counterexample as appropriate.
(i) If and are continuous then is continuous.
(ii) If is strictly increasing and is continuous then is continuous.
(iii) If is continuous and a bijection then is continuous.
(iv) If is differentiable and a bijection then is differentiable.
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Paper 1, Section II, F
2021 commentLet be a continuous function.
(a) Let and . If is a positive continuous function, prove that
directly from the definition of the Riemann integral.
(b) Let be a continuous function. Show that
as , and deduce that
as
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Paper 1, Section II, F
2021 commentLet be -times differentiable, for some .
(a) State and prove Taylor's theorem for , with the Lagrange form of the remainder. [You may assume Rolle's theorem.]
(b) Suppose that is an infinitely differentiable function such that and , and satisfying the differential equation . Prove carefully that
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Paper 1, Section II, F
2021 comment(a) Let be a power series with . Show that there exists (called the radius of convergence) such that the series is absolutely convergent when but is divergent when .
Suppose that the radius of convergence of the series is . For a fixed positive integer , find the radii of convergence of the following series. [You may assume that exists.] (i) . (ii) . (iii) .
(b) Suppose that there exist values of for which converges and values for which it diverges. Show that there exists a real number such that diverges whenever and converges whenever .
Determine the set of values of for which
converges.
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Paper 2, Section I, A
2021 commentSolve the difference equation
subject to the initial conditions and .
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Paper 2, Section I, A
2021 commentLet and be two linearly independent solutions to the differential equation
Show that the Wronskian satisfies
Deduce that if then
Given that satisfies the equation
find the solution which satisfies and .
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Paper 2, Section II, A
2021 commentThe function takes values in the interval and satisfies the differential equation
where and are positive constants.
Let . Express in terms of a pair of first order differential equations in . Show that if then there are three fixed points in the region
Classify all the fixed points of the system in the case . Sketch the phase portrait in the case and .
Comment briefly on the case when .
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Paper 2, Section II, A
2021 commentFor a linear, second order differential equation define the terms ordinary point, singular point and regular singular point.
For and consider the following differential equation
Find coefficients such that the function , where
satisfies . By making the substitution , or otherwise, find a second linearly independent solution of the form for suitable .
Suppose now that . By considering a limit of the form
or otherwise, obtain two linearly independent solutions to in terms of and derivatives thereof.
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Paper 2, Section II, A
2021 commentBy means of the change of variables and , show that the wave equation for
is equivalent to the equation
where . Hence show that the solution to on and , subject to the initial conditions
Deduce that if and on the interval then on .
Suppose now that is a solution to the wave equation on the finite interval and obeys the boundary conditions
for all . The energy is defined by
By considering , or otherwise, show that the energy remains constant in time.
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Paper 2, Section II, A
2021 commentFor an matrix , define the matrix exponential by
where , with being the identity matrix. [You may assume that for real numbers and you do not need to consider issues of convergence.] Show that
Deduce that the unique solution to the initial value problem
is .
Let and be vectors of length and a real matrix. By considering a suitable integrating factor, show that the unique solution to
is given by
Hence, or otherwise, solve the system of differential equations when
[Hint: Compute and show that
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Paper 4, Section I, C
2021 commentA trolley travels with initial speed along a frictionless, horizontal, linear track. It slows down by ejecting gas in the direction of motion. The gas is emitted at a constant mass ejection rate and with constant speed relative to the trolley. The trolley and its supply of gas initially have a combined mass of . How much time is spent ejecting gas before the trolley stops? [Assume that the trolley carries sufficient gas.]
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Paper 4, Section I, C
2021 commentA rigid body composed of particles with positions , and masses , rotates about the -axis with constant angular speed . Show that the body's kinetic energy is , where you should give an expression for the moment of inertia in terms of the particle masses and positions.
Consider a solid cuboid of uniform density, mass , and dimensions . Choose coordinate axes so that the cuboid is described by the points with , and . In terms of , , and , find the cuboid's moment of inertia for rotations about the -axis.
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Paper 4, Section II, C
2021 commentA particle of mass follows a one-dimensional trajectory in the presence of a variable force . Write down an expression for the work done by this force as the particle moves from to . Assuming that this is the only force acting on the particle, show that the work done by the force is equal to the change in the particle's kinetic energy.
What does it mean if a force is said to be conservative?
A particle moves in a force field given by
where and are positive constants. The particle starts at the origin with initial velocity . Show that, as the particle's position increases from to larger , the particle's velocity at position is given by
where you should determine . What determines whether the particle will escape to infinity or oscillate about the origin? Sketch versus for each of these cases, carefully identifying any significant velocities or positions.
In the case of oscillatory motion, find the period of oscillation in terms of , and . [Hint: You may use the fact that
for .]
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Paper 4, Section II, 10C
2021 comment(a) A mass is acted upon by a central force
where is a positive constant and is the displacement of the mass from the origin. Show that the angular momentum and energy of the mass are conserved.
(b) Working in plane polar coordinates , or otherwise, show that the distance between the mass and the origin obeys the following differential equation
where is the angular momentum per unit mass.
(c) A satellite is initially in a circular orbit of radius and experiences the force described above. At and time , the satellite emits a short rocket burst putting it on an elliptical orbit with its closest distance to the centre and farthest distance . When and the time is , the satellite reaches the farthest distance and a second short rocket burst puts the rocket on a circular orbit of radius . (See figure.) [Assume that the duration of the rocket bursts is negligible.]

(i) Show that the satellite's angular momentum per unit mass while in the elliptical orbit is
where is a number you should determine.
(ii) What is the change in speed as a result of the rocket burst at time ? And what is the change in speed at ?
(iii) Given that the elliptical orbit can be described by
where is the eccentricity of the orbit, find in terms of , and . [Hint: The area of an ellipse is equal to , where and b are its semi-major and semi-minor axes; these are related to the eccentricity by
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Paper 4, Section II, C
2021 commentConsider an inertial frame of reference and a frame of reference which is rotating with constant angular velocity relative to . Assume that the two frames have a common origin .
Let be any vector. Explain why the derivative of in frame is related to its derivative in by the following equation
[Hint: It may be useful to use Cartesian basis vectors in both frames.]
Let be the position vector of a particle, measured from . Derive the expression relating the particle's acceleration as observed in , to the acceleration observed in , written in terms of and
A small bead of mass is threaded on a smooth, rigid, circular wire of radius . At any given instant, the wire hangs in a vertical plane with respect to a downward gravitational acceleration . The wire is rotating with constant angular velocity about its vertical diameter. Let be the angle between the downward vertical and the radial line going from the centre of the hoop to the bead.
(i) Show that satisfies the following equation of motion
(ii) Find any equilibrium angles and determine their stability.
(iii) Find the force of the wire on the bead as a function of and .
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Paper 4, Section II, C
2021 commentWrite down the expression for the momentum of a particle of rest mass , moving with velocity where is near the speed of light . Write down the corresponding 4-momentum.
Such a particle experiences a force . Why is the following expression for the particle's acceleration,
not generally correct? Show that the force can be written as follows
Invert this expression to find the particle's acceleration as the sum of two vectors, one parallel to and one parallel to .
A particle with rest mass and charge is in the presence of a constant electric field which exerts a force on the particle. If the particle is at rest at , its motion will be in the direction of for . Determine the particle's speed for . How does the velocity behave as ?
[Hint: You may find that trigonometric substitution is helpful in evaluating an integral.]
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Paper 3 , Section I, D
2021 commentLet be a finite group and denote the centre of by . Prove that if the quotient group is cyclic then is abelian. Does there exist a group such that (i) ? (ii) ?
Justify your answers.
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Paper 3, Section I, D
2021 commentLet and be elements of a group . What does it mean to say and are conjugate in ? Prove that if two elements in a group are conjugate then they have the same order.
Define the Möbius group . Prove that if are conjugate they have the same number of fixed points. Quoting clearly any results you use, show that any nontrivial element of of finite order has precisely 2 fixed points.
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Paper 3, Section II, D
2021 comment(a) Let be an element of a finite group . Define the order of and the order of . State and prove Lagrange's theorem. Deduce that the order of divides the order of .
(b) If is a group of order , and is a divisor of where , is it always true that must contain an element of order ? Justify your answer.
(c) Denote the cyclic group of order by .
(i) Prove that if and are coprime then the direct product is cyclic.
(ii) Show that if a finite group has all non-identity elements of order 2 , then is isomorphic to . [The direct product theorem may be used without proof.]
(d) Let be a finite group and a subgroup of .
(i) Let be an element of order in . If is the least positive integer such that , show that divides .
(ii) Suppose further that has index . If , show that for some such that . Is it always the case that the least positive such is a factor of ? Justify your answer.
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Paper 3, Section II, D
2021 comment(a) Let be a finite group acting on a set . For , define the orbit and the stabiliser of . Show that is a subgroup of . State and prove the orbit-stabiliser theorem.
(b) Let be integers. Let , the symmetric group of degree , and be the set of all ordered -tuples with . Then acts on , where the action is defined by for and . For , determine and and verify that the orbit-stabiliser theorem holds in this case.
(c) We say that acts doubly transitively on if, whenever and are elements of with and , there exists some such that and .
Assume that is a finite group that acts doubly transitively on , and let . Show that if is a subgroup of that properly contains that is, but then the action of on is transitive. Deduce that .
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Paper 3, Section II,
2021 commentLet be a finite group of order . Show that is isomorphic to a subgroup of , the symmetric group of degree . Furthermore show that this isomorphism can be chosen so that any nontrivial element of has no fixed points.
Suppose is even. Prove that contains an element of order 2 .
What does it mean for an element of to be odd? Suppose is a subgroup of for some , and contains an odd element. Prove that precisely half of the elements of are odd.
Now suppose for some positive integer . Prove that is not simple. [Hint: Consider the sign of an element of order 2.]
Can a nonabelian group of even order be simple?
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Paper 3, Section II, D
2021 comment(a) Let be an abelian group (not necessarily finite). We define the generalised dihedral group to be the set of pairs
with multiplication given by
The identity is and the inverse of is . You may assume that this multiplication defines a group operation on .
(i) Identify with the set of all pairs in which . Show that is a subgroup of . By considering the index of in , or otherwise, show that is a normal subgroup of .
(ii) Show that every element of not in has order 2 . Show that is abelian if and only if for all . If is non-abelian, what is the centre of Justify your answer.
(b) Let denote the group of orthogonal matrices. Show that all elements of have determinant 1 or . Show that every element of is a rotation. Let . Show that decomposes as a union .
[You may assume standard properties of determinants.]
(c) Let be the (abelian) group , with multiplication of complex numbers as the group operation. Write down, without proof, isomorphisms where denotes the additive group of real numbers and the subgroup of integers. Deduce that , the generalised dihedral group defined in part (a).
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Paper 4 , Section I, E
2021 commentConsider functions and . Which of the following statements are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) If is surjective then is surjective.
(ii) If is injective then is injective.
(iii) If is injective then is injective.
If and with , and is the identity on , then how many possibilities are there for the pair of functions and ?
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Paper 4, Section I,
2021 commentThe Fibonacci numbers are defined by . Let be the ratio of successive Fibonacci numbers.
(i) Show that . Hence prove by induction that
for all . Deduce that the sequence is monotonically decreasing.
(ii) Prove that
for all . Hence show that as .
(iii) Explain without detailed justification why the sequence has a limit.
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Paper 4, Section II, E
2021 comment(a) Let be the set of all functions . Define by
(i) Define the binomial coefficient for . Setting when , prove from your definition that if then .
(ii) Show that if is integer-valued and , then
for some integers .
(b) State the binomial theorem. Show that
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Paper 4, Section II,
2021 comment(a) (i) By considering Euclid's algorithm, show that the highest common factor of two positive integers and can be written in the form for suitable integers and . Find an integer solution of
Is your solution unique?
(ii) Suppose that and are coprime. Show that the simultaneous congruences
have the same set of solutions as for some . Hence solve (i.e. find all solutions of) the simultaneous congruences
(b) State the inclusion-exclusion principle.
For integers , denote by the number of ordered r-tuples of integers satisfying for and such that the greatest common divisor of is 1 . Show that
where the product is over all prime numbers dividing .
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Paper 4, Section II,
2021 comment(a) Prove that every real number can be written in the form where is a strictly increasing sequence of positive integers.
Are such expressions unique?
(b) Let be a root of , where . Suppose that has no rational roots, except possibly .
(i) Show that if then
where is a constant depending only on .
(ii) Deduce that if with and then
(c) Prove that is transcendental.
(d) Let and be transcendental numbers. What of the following statements are always true and which can be false? Briefly justify your answers.
(i) is transcendental.
(ii) is transcendental for every .
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Paper 4, Section II, 8E
2021 comment(a) Prove that a countable union of countable sets is countable.
(b) (i) Show that the set of all functions is uncountable.
(ii) Determine the countability or otherwise of each of the two sets
Justify your answers.
(c) A permutation of the natural numbers is a mapping that is bijective. Determine the countability or otherwise of each of the two sets and of permutations, justifying your answers:
(i) is the set of all permutations of such that for all sufficiently large .
(ii) is the set all permutations of such that
for each .
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Paper 2, Section I, D
2021 commentA coin has probability of landing heads. Let be the probability that the number of heads after tosses is even. Give an expression for in terms of . Hence, or otherwise, find .
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Paper 2, Section I, F
2021 commentLet be a continuous random variable taking values in . Let the probability density function of be
where is a constant.
Find the value of and calculate the mean, variance and median of .
[Recall that the median of is the number such that
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Paper 2, Section II, 9E
2021 comment(a) (i) Define the conditional probability of the event given the event . Let be a partition of the sample space such that for all . Show that, if ,
(ii) There are urns, the th of which contains red balls and blue balls. Alice picks an urn (uniformly) at random and removes two balls without replacement. Find the probability that the first ball is blue, and the conditional probability that the second ball is blue, given that the first is blue. [You may assume, if you wish, that .]
(b) (i) What is meant by saying that two events and are independent? Two fair (6-sided) dice are rolled. Let be the event that the sum of the numbers shown is , and let be the event that the first die shows . For what values of and are the two events and independent?
(ii) The casino at Monte Corona features the following game: three coins each show heads with probability and tails otherwise. The first counts 10 points for a head and 2 for a tail; the second counts 4 points for both a head and a tail; and the third counts 3 points for a head and 20 for a tail. You and your opponent each choose a coin. You cannot both choose the same coin. Each of you tosses your coin once and the person with the larger score wins the jackpot. Would you prefer to be the first or the second to choose a coin?
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Paper 2, Section II, 10E
2021 comment(a) Alanya repeatedly rolls a fair six-sided die. What is the probability that the first number she rolls is a 1 , given that she rolls a 1 before she rolls a
(b) Let be a simple symmetric random walk on the integers starting at , that is,
where is a sequence of IID random variables with . Let be the time that the walk first hits 0 .
(i) Let be a positive integer. For , calculate the probability that the walk hits 0 before it hits .
(ii) Let and let be the event that the walk hits 0 before it hits 3 . Find . Hence find .
(iii) Let and let be the event that the walk hits 0 before it hits 4 . Find .
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Paper 2, Section II, D
2021 commentLet be the disc of radius 1 with centre at the origin . Let be a random point uniformly distributed in . Let be the polar coordinates of . Show that and are independent and find their probability density functions and .
Let and be three random points selected independently and uniformly in . Find the expected area of triangle and hence find the probability that lies in the interior of triangle .
Find the probability that and are the vertices of a convex quadrilateral.
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Paper 2, Section II, 12F
2021 commentState and prove Chebyshev's inequality.
Let be a sequence of independent, identically distributed random variables such that
for some , and let be a continuous function.
(i) Prove that
is a polynomial function of , for any natural number .
(ii) Let . Prove that
where is the set of natural numbers such that .
(iii) Show that
as . [You may use without proof that, for any , there is a such that for all with .]
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Paper 3, Section I, B
2021 comment(a) Prove that
where and are differentiable vector fields and is a differentiable scalar field.
(b) Find the solution of on the two-dimensional domain when
(i) is the unit disc , and on ;
(ii) is the annulus , and on both and .
[Hint: the Laplacian in plane polar coordinates is:
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Paper 3, Section I, B
2021 comment(a) What is meant by an antisymmetric tensor of second rank? Show that if a second rank tensor is antisymmetric in one Cartesian coordinate system, it is antisymmetric in every Cartesian coordinate system.
(b) Consider the vector field and the second rank tensor defined by . Calculate the components of the antisymmetric part of and verify that it equals , where is the alternating tensor and .
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Paper 3, Section II, B
2021 comment(a) Given a space curve , with a parameter (not necessarily arc-length), give mathematical expressions for the unit tangent, unit normal, and unit binormal vectors.
(b) Consider the closed curve given by
where .
Show that the unit tangent vector may be written as
with each sign associated with a certain range of , which you should specify.
Calculate the unit normal and the unit binormal vectors, and hence deduce that the curve lies in a plane.
(c) A closed space curve lies in a plane with unit normal . Use Stokes' theorem to prove that the planar area enclosed by is the absolute value of the line integral
Hence show that the planar area enclosed by the curve given by is .
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Paper 3, Section II, B
2021 comment(a) By considering an appropriate double integral, show that
where .
(b) Calculate , treating as a constant, and hence show that
(c) Consider the region in the plane enclosed by , and with .
Sketch , indicating any relevant polar angles.
A surface is given by . Calculate the volume below this surface and above .
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Paper 3, Section II, B
2021 comment(a) By a suitable change of variables, calculate the volume enclosed by the ellipsoid , where , and are constants.
(b) Suppose is a second rank tensor. Use the divergence theorem to show that
where is a closed surface, with unit normal , and is the volume it encloses.
[Hint: Consider for a constant vector
(c) A half-ellipsoidal membrane is described by the open surface , with . At a given instant, air flows beneath the membrane with velocity , where is a constant. The flow exerts a force on the membrane given by
where is a constant parameter.
Show the vector can be rewritten as .
Hence use to calculate the force on the membrane.
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Paper 3, Section II, B
2021 commentFor a given charge distribution and current distribution in , the electric and magnetic fields, and , satisfy Maxwell's equations, which in suitable units, read
The Poynting vector is defined as .
(a) For a closed surface around a volume , show that
(b) Suppose and consider an electromagnetic wave
where and are positive constants. Show that these fields satisfy Maxwell's equations for appropriate , and .
Confirm the wave satisfies the integral identity by considering its propagation through a box , defined by , and .
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Paper 1, Section I, C
2021 comment(a) Find all complex solutions to the equation .
(b) Write down an equation for the numbers which describe, in the complex plane, a circle with radius 5 centred at . Find the points on the circle at which it intersects the line passing through and .
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Paper 1, Section I, B
2021 commentThe matrix
represents a linear map with respect to the bases
Find the matrix that represents with respect to the bases
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Paper 1, Section II, C
2021 commentUsing the standard formula relating products of the Levi-Civita symbol to products of the Kronecker , prove
Define the scalar triple product of three vectors , and in in terms of the dot and cross product. Show that
Given a basis for which is not necessarily orthonormal, let
Show that is also a basis for . [You may assume that three linearly independent vectors in form a basis.]
The vectors are constructed from in the same way that , are constructed from . Show that
An infinite lattice consists of all points with position vectors given by
Find all points with position vectors such that is an integer for all integers , .
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Paper 1, Section II, A
2021 comment(a) For an matrix define the characteristic polynomial and the characteristic equation.
The Cayley-Hamilton theorem states that every matrix satisfies its own characteristic equation. Verify this in the case .
(b) Define the adjugate matrix of an matrix in terms of the minors of . You may assume that
where is the identity matrix. Show that if and are non-singular matrices then
(c) Let be an arbitrary matrix. Explain why
(i) there is an such that is non-singular for ;
(ii) the entries of are polynomials in .
Using parts (i) and (ii), or otherwise, show that holds for all matrices .
(d) The characteristic polynomial of the arbitrary matrix is
By considering adj , or otherwise, show that
[You may assume the Cayley-Hamilton theorem.]
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Paper 1, Section II, A
2021 commentLet be a real, symmetric matrix.
We say that is positive semi-definite if for all . Prove that is positive semi-definite if and only if all the eigenvalues of are non-negative. [You may quote results from the course, provided that they are clearly stated.]
We say that has a principal square root if for some symmetric, positive semi-definite matrix . If such a exists we write . Show that if is positive semi-definite then exists.
Let be a real, non-singular matrix. Show that is symmetric and positive semi-definite. Deduce that exists and is non-singular. By considering the matrix
or otherwise, show for some orthogonal matrix and a symmetric, positive semi-definite matrix .
Describe the transformation geometrically in the case .
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Paper 1, Section II, 8B
2021 comment(a) Consider the matrix
Find the kernel of for each real value of the constant . Hence find how many solutions there are to
depending on the value of . [There is no need to find expressions for the solution(s).]
(b) Consider the reflection map defined as
where is a unit vector normal to the plane of reflection.
(i) Find the matrix which corresponds to the map in terms of the components of .
(ii) Prove that a reflection in a plane with unit normal followed by a reflection in a plane with unit normal vector (both containing the origin) is equivalent to a rotation along the line of intersection of the planes with an angle twice that between the planes.
[Hint: Choose your coordinate axes carefully.]
(iii) Briefly explain why a rotation followed by a reflection or vice-versa can never be equivalent to another rotation.
Part IA, 2021 List of Questions
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Paper 2, Section I,
2021 commentLet be a continuous function and let denote the set of continuous real-valued functions on . Given , define the function by the expression
(a) Prove that is a continuous map with the uniform metric on .
(b) Let be the metric on given by
Is continuous with respect to
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Paper 4, Section I,
2021 commentLet be a topological space with an equivalence relation, the set of equivalence classes, , the quotient map taking a point in to its equivalence class.
(a) Define the quotient topology on and check it is a topology.
(b) Prove that if is a topological space, a map is continuous if and only if is continuous.
(c) If is Hausdorff, is it true that is also Hausdorff? Justify your answer.
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Paper 1, Section II, F
2021 commentLet be a map between metric spaces. Prove that the following two statements are equivalent:
(i) is open whenever is open.
(ii) for any sequence .
For as above, determine which of the following statements are always true and which may be false, giving a proof or a counterexample as appropriate.
(a) If is compact and is continuous, then is uniformly continuous.
(b) If is compact and is continuous, then is compact.
(c) If is connected, is continuous and is dense in , then is connected.
(d) If the set is closed in and is compact, then is continuous.
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Paper 2, Section II, F
2021 commentLet be a sequence of functions satisfying the following properties:
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for all and and there is such that vanishes outside for all
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each is continuous and
- given and , there exists a positive integer such that if , then
Let be a bounded continuous function and set
Show that converges uniformly to on any compact subset of .
Let be a continuous function with . Show that there is a sequence of polynomials such that converges uniformly to on . Hint: consider the functions
where is a suitably chosen constant.]
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Paper 3, Section II, F
2021 commentDefine the terms connected and path-connected for a topological space. Prove that the interval is connected and that if a topological space is path-connected, then it is connected.
Let be an open subset of Euclidean space . Show that is connected if and only if is path-connected.
Let be a topological space with the property that every point has a neighbourhood homeomorphic to an open set in . Assume is connected; must be also pathconnected? Briefly justify your answer.
Consider the following subsets of :
Let
with the subspace topology. Is path-connected? Is connected? Justify your answers.
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Paper 4, Section II, F
2021 comment(a) Let be a continuous function such that for each , the partial derivatives of exist and are continuous on . Define by
Show that has continuous partial derivatives given by
for .
(b) Let be an infinitely differentiable function, that is, partial derivatives exist and are continuous for all and . Show that for any ,
where is an infinitely differentiable function.
[Hint: You may use the fact that if is infinitely differentiable, then
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Paper 4, Section I,
2021 commentLet be a holomorphic function on a neighbourhood of . Assume that has a zero of order at with . Show that there exist and such that for any with there are exactly distinct values of with .
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Paper 3, Section II, G
2021 commentLet be a curve (not necessarily closed) in and let denote the image of . Let be a continuous function and define
for . Show that has a power series expansion about every .
Using Cauchy's Integral Formula, show that a holomorphic function has complex derivatives of all orders. [Properties of power series may be assumed without proof.] Let be a holomorphic function on an open set that contains the closed disc . Obtain an integral formula for the derivative of on the open disc in terms of the values of on the boundary of the disc.
Show that if holomorphic functions on an open set converge locally uniformly to a holomorphic function on , then converges locally uniformly to .
Let and be two overlapping closed discs. Let be a holomorphic function on some open neighbourhood of . Show that there exist open neighbourhoods of and holomorphic functions on , such that on .
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Paper 1, Section I, B
2021 commentLet , and let denote the positively oriented circle of radius centred at the origin. Define
Evaluate for .
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Paper 1, Section II, G
2021 comment(a) State a theorem establishing Laurent series of analytic functions on suitable domains. Give a formula for the Laurent coefficient.
Define the notion of isolated singularity. State the classification of an isolated singularity in terms of Laurent coefficients.
Compute the Laurent series of
on the annuli and . Using this example, comment on the statement that Laurent coefficients are unique. Classify the singularity of at 0 .
(b) Let be an open subset of the complex plane, let and let . Assume that is an analytic function on with as . By considering the Laurent series of at , classify the singularity of at in terms of the Laurent coefficients. [You may assume that a continuous function on that is analytic on is analytic on .]
Now let be an entire function with as . By considering Laurent series at 0 of and of , show that is a polynomial.
(c) Classify, giving reasons, the singularity at the origin of each of the following functions and in each case compute the residue:
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Paper 2, Section II, B
2021 comment(a) Let be an entire function and let be constants. Show that if
for all , where is a positive odd integer, then must be a polynomial with degree not exceeding (closest integer part rounding down).
Does there exist a function , analytic in , such that for all nonzero Justify your answer.
(b) State Liouville's Theorem and use it to show the following.
(i) If is a positive harmonic function on , then is a constant function.
(ii) Let be a line in where . If is an entire function such that , then is a constant function.
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Paper 3, Section I, B
2021 commentFind the value of for which the function
satisfies Laplace's equation. For this value of , find a complex analytic function of which is the real part.
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Paper 4, Section II, B
2021 commentLet be defined for . Define the Laplace transform of . Find an expression for the Laplace transform of in terms of .
Three radioactive nuclei decay sequentially, so that the numbers of the three types obey the equations
where are constants. Initially, at and . Using Laplace transforms, find .
By taking an appropriate limit, find when and .
-
Paper 2, Section I,
2021 commentState Gauss's Law in the context of electrostatics.
A simple coaxial cable consists of an inner conductor in the form of a perfectly conducting, solid cylinder of radius , surrounded by an outer conductor in the form of a perfectly conducting, cylindrical shell of inner radius and outer radius . The cylinders are coaxial and the gap between them is filled with a perfectly insulating material. The cable may be assumed to be straight and arbitrarily long.
In a steady state, the inner conductor carries an electric charge per unit length, and the outer conductor carries an electric charge per unit length. The charges are distributed in a cylindrically symmetric way and no current flows through the cable.
Determine the electrostatic potential and the electric field as functions of the cylindrical radius , for . Calculate the capacitance of the cable per unit length and the electrostatic energy per unit length, and verify that these are related by
-
Paper 4, Section I,
2021 commentWrite down Maxwell's equations in a vacuum. Show that they admit wave solutions with
where and must obey certain conditions that you should determine. Find the corresponding electric field .
A light wave, travelling in the -direction and linearly polarised so that the magnetic field points in the -direction, is incident upon a conductor that occupies the half-space . The electric and magnetic fields obey the boundary conditions and on the surface of the conductor, where is the unit normal vector. Determine the contributions to the magnetic field from the incident and reflected waves in the region . Compute the magnetic field tangential to the surface of the conductor.
-
Paper 1, Section II, 15D
2021 comment(a) Show that the magnetic flux passing through a simple, closed curve can be written as
where is the magnetic vector potential. Explain why this integral is independent of the choice of gauge.
(b) Show that the magnetic vector potential due to a static electric current density , in the Coulomb gauge, satisfies Poisson's equation
Hence obtain an expression for the magnetic vector potential due to a static, thin wire, in the form of a simple, closed curve , that carries an electric current . [You may assume that the electric current density of the wire can be written as
where is the three-dimensional Dirac delta function.]
(c) Consider two thin wires, in the form of simple, closed curves and , that carry electric currents and , respectively. Let (where ) be the magnetic flux passing through the curve due to the current flowing around . The inductances are defined by . By combining the results of parts (a) and (b), or otherwise, derive Neumann's formula for the mutual inductance,
Suppose that is a circular loop of radius , centred at and lying in the plane , and that is a different circular loop of radius , centred at and lying in the plane . Show that the mutual inductance of the two loops is
where
and the function is defined, for , by the integral
-
Paper 2, Section II,
2021 comment(a) Show that, for ,
(b) A particle with electric charge has position vector , where . An earthed conductor (held at zero potential) occupies the plane . Explain why the boundary conditions can be satisfied by introducing a fictitious 'image' particle of appropriate charge and position. Hence determine the electrostatic potential and the electric field in the region . Find the leading-order approximation to the potential for and compare with that of an electric dipole. Directly calculate the total flux of the electric field through the plane and comment on the result. Find the induced charge distribution on the surface of the conductor, and the total induced surface charge. Sketch the electric field lines in the plane .
(c) Now consider instead a particle with charge at position , where and , with earthed conductors occupying the planes and . Find the leading-order approximation to the potential in the region for and state what type of multipole potential this is.
-
Paper 3, Section II, 15D
2021 comment(a) The energy density stored in the electric and magnetic fields and is given by
Show that, in regions where no electric current flows,
for some vector field that you should determine.
(b) The coordinates in an inertial frame are related to the coordinates in an inertial frame by a Lorentz transformation , where
with . Here is the relative velocity of with respect to in the x-direction.
In frame , there is a static electric field with , and no magnetic field. Calculate the electric field and magnetic field in frame . Show that the energy density in frame is given in terms of the components of by
Use the fact that to show that
where is the unit vector in the -direction.
-
Paper 2, Section I, A
2021 commentConsider an axisymmetric container, initially filled with water to a depth . A small circular hole of radius is opened in the base of the container at .
(a) Determine how the radius of the container should vary with so that the depth of the water will decrease at a constant rate.
(b) For such a container, determine how the cross-sectional area of the free surface should decrease with time.
[You may assume that the flow rate through the opening is sufficiently small that Bernoulli's theorem for steady flows can be applied.]
-
Paper 3, Section I, A
2021 commentA two-dimensional flow has a velocity field given by
(a) Show explicitly that this flow is incompressible and irrotational away from the origin.
(b) Find the stream function for this flow.
(c) Find the velocity potential for this flow.
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Paper 1, Section II, A
2021 commentA two-dimensional flow is given by a velocity potential
where is a constant.
(a) Find the corresponding velocity field . Determine .
(b) The time-average of a quantity is defined as
Show that the time-average of this velocity field is zero everywhere. Write down an expression for the acceleration of fluid particles, and find the time-average of this expression at a fixed point .
(c) Now assume that . The material particle at at is marked with dye. Write down equations for its subsequent motion. Verify that its position for is given (correct to terms of order ) by
Deduce the time-average velocity of the dyed particle correct to this order.
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Paper 3, Section II, A
2021 commentA two-dimensional layer of viscous fluid lies between two rigid boundaries at . The boundary at oscillates in its own plane with velocity , while the boundary at oscillates in its own plane with velocity . Assume that there is no pressure gradient and that the fluid flows parallel to the boundary with velocity , where can be written as .
(a) By exploiting the symmetry of the system or otherwise, show that
(b) Hence or otherwise, show that
where .
(c) Show that, for ,
and briefly interpret this result physically.
-
Paper 4, Section II, A
2021 commentConsider the spherically symmetric motion induced by the collapse of a spherical cavity of radius , centred on the origin. For , there is a vacuum, while for , there is an inviscid incompressible fluid with constant density . At time , and the fluid is at rest and at constant pressure .
(a) Consider the radial volume transport in the fluid , defined as
where is the radial velocity, and is an infinitesimal element of the surface of a sphere of radius . Use the incompressibility condition to establish that is a function of time alone.
(b) Using the expression for pressure in potential flow or otherwise, establish that
where is the radial velocity of the cavity boundary.
(c) By expressing in terms of and , show that
[Hint: You may find it useful to assume is an explicit function of a from the outset.]
(d) Hence write down an integral expression for the implosion time , i.e. the time for the radius of the cavity . [Do not attempt to evaluate the integral.]
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Paper 1, Section I, F
2021 commentLet be a smooth function and let (assumed not empty). Show that if the differential for all , then is a smooth surface in .
Is a smooth surface? Is every surface of the form for some smooth ? Justify your answers.
-
Paper 3, Section I, E
2021 commentState the local Gauss-Bonnet theorem for geodesic triangles on a surface. Deduce the Gauss-Bonnet theorem for closed surfaces. [Existence of a geodesic triangulation can be assumed.]
Let denote the sphere with radius centred at the origin. Show that the Gauss curvature of is . An octant is any of the eight regions in bounded by arcs of great circles arising from the planes . Verify directly that the local Gauss-Bonnet theorem holds for an octant. [You may assume that the great circles on are geodesics.]
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Paper 1, Section II, F
2021 commentLet be an oriented surface. Define the Gauss map and show that the differential of the Gauss map at any point is a self-adjoint linear map. Define the Gauss curvature and compute in a given parametrisation.
A point is called umbilic if has a repeated eigenvalue. Let be a surface such that every point is umbilic and there is a parametrisation such that . Prove that is part of a plane or part of a sphere. Hint: consider the symmetry of the mixed partial derivatives , where for
-
Paper 2, Section II, E
2021 commentDefine , the upper half plane model for the hyperbolic plane, and show that acts on by isometries, and that these isometries preserve the orientation of .
Show that every orientation preserving isometry of is in , and hence the full group of isometries of is , where .
Let be a hyperbolic line. Define the reflection in . Now let be two hyperbolic lines which meet at a point at an angle . What are the possibilities for the group generated by and ? Carefully justify your answer.
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Paper 3, Section II, E
2021 commentLet be an embedded smooth surface and a parameterised smooth curve on . What is the energy of ? By applying the Euler-Lagrange equations for stationary curves to the energy function, determine the differential equations for geodesics on explicitly in terms of a parameterisation of .
If contains a straight line , prove from first principles that each segment (with some parameterisation) is a geodesic on .
Let be the hyperboloid defined by the equation and let . By considering appropriate isometries, or otherwise, display explicitly three distinct (as subsets of ) geodesics through in the case when and four distinct geodesics through in the case when . Justify your answer.
Let be a geodesic, with coordinates . Clairaut's relation asserts is constant, where and is the angle between and the plane through the point and the -axis. Deduce from Clairaut's relation that there exist infinitely many geodesics on which stay in the half-space for all .
[You may assume that if satisfies the geodesic equations on then is defined for all and the Euclidean norm is constant. If you use a version of the geodesic equations for a surface of revolution, then that should be proved.]
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Paper 4, Section II, F
2021 commentDefine an abstract smooth surface and explain what it means for the surface to be orientable. Given two smooth surfaces and and a map , explain what it means for to be smooth
For the cylinder
let be the orientation reversing diffeomorphism . Let be the quotient of by the equivalence relation and let be the canonical projection map. Show that can be made into an abstract smooth surface so that is smooth. Is orientable? Justify your answer.
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Paper 2, Section I,
2021 commentLet be a module over a Principal Ideal Domain and let be a submodule of . Show that is finitely generated if and only if and are finitely generated.
-
Paper 3, Section I, G
2021 commentLet be a finite group, and let be a proper subgroup of of index .
Show that there is a normal subgroup of such that divides ! and .
Show that if is non-abelian and simple, then is isomorphic to a subgroup of .
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Paper 1, Section II, G
2021 commentShow that a ring is Noetherian if and only if every ideal of is finitely generated. Show that if is a surjective ring homomorphism and is Noetherian, then is Noetherian.
State and prove Hilbert's Basis Theorem.
Let . Is Noetherian? Justify your answer.
Give, with proof, an example of a Unique Factorization Domain that is not Noetherian.
Let be the ring of continuous functions . Is Noetherian? Justify your answer.
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Paper 2, Section II, G
2021 commentLet be a module over a ring and let . Define what it means that freely generates . Show that this happens if and only if for every -module , every function extends uniquely to a homomorphism .
Let be a free module over a (non-trivial) ring that is generated (not necessarily freely) by a subset of size . Show that if is a basis of , then is finite with . Hence, or otherwise, deduce that any two bases of have the same number of elements. Denoting this number and by quoting any result you need, show that if is a Euclidean Domain and is a submodule of , then is free with .
State the Primary Decomposition Theorem for a finitely generated module over a Euclidean Domain . Deduce that any finite subgroup of the multiplicative group of a field is cyclic.
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Paper 3, Section II, 10G
2021 commentLet be a non-zero element of a Principal Ideal Domain . Show that the following are equivalent:
(i) is prime;
(ii) is irreducible;
(iii) is a maximal ideal of ;
(iv) is a field;
(v) is an Integral Domain.
Let be a Principal Ideal Domain, an Integral Domain and a surjective ring homomorphism. Show that either is an isomorphism or is a field.
Show that if is a commutative ring and is a Principal Ideal Domain, then is a field.
Let be an Integral Domain in which every two non-zero elements have a highest common factor. Show that in every irreducible element is prime.
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Paper 4, Section II, G
2021 commentLet and be subgroups of a finite group . Show that the sets , partition . By considering the action of on the set of left cosets of in by left multiplication, or otherwise, show that
for any . Deduce that if has a Sylow -subgroup, then so does .
Let with a prime. Write down the order of the group . Identify in a Sylow -subgroup and a subgroup isomorphic to the symmetric group . Deduce that every finite group has a Sylow -subgroup.
State Sylow's theorem on the number of Sylow -subgroups of a finite group.
Let be a group of order , where are prime numbers. Show that if is non-abelian, then .
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Paper 1, Section I,
2021 commentLet be a vector space over , and let , symmetric bilinear form on .
Let . Show that is of dimension and . Show that if is a subspace with , then the restriction of , is nondegenerate.
Conclude that the dimension of is even.
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Paper 4, Section I,
2021 commentLet be the vector space of by complex matrices.
Given , define the linear ,
(i) Compute a basis of eigenvectors, and their associated eigenvalues, when is the diagonal matrix
What is the rank of ?
(ii) Now let . Write down the matrix of the linear transformation with respect to the standard basis of .
What is its Jordan normal form?
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Paper 1, Section II, E
2021 commentLet , and let .
(a) (i) Compute , for all .
(ii) Hence, or otherwise, compute , for all .
(b) Let be a finite-dimensional vector space over , and let . Suppose for some .
(i) Determine the possible eigenvalues of .
(ii) What are the possible Jordan blocks of ?
(iii) Show that if , there exists a decomposition
where , and .
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Paper 2, Section II, E
2021 comment(a) Compute the characteristic polynomial and minimal polynomial of
Write down the Jordan normal form for .
(b) Let be a finite-dimensional vector space over be a linear map, and for , write
(i) Given , construct a non-zero eigenvector for in terms of .
(ii) Show that if are non-zero eigenvectors for with eigenvalues , and for all , then are linearly independent.
(iii) Show that if are all non-zero, and for all , then are linearly independent.
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Paper 3, Section II, 9E
2021 comment(a) (i) State the rank-nullity theorem.
Let and be vector spaces. Write down the definition of their direct sum and the inclusions .
Now let and be subspaces of a vector space . Define by
Describe the quotient space as a subspace of .
(ii) Let , and let be the subspace of spanned by the vectors
and the subspace of spanned by the vectors
Determine the dimension of .
(b) Let be complex by matrices with .
Show that is a polynomial in of degree at most .
Show that if the polynomial is of degree precisely .
Give an example where but this polynomial is zero.
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Paper 4, Section II, E
2021 comment(a) Let be a complex vector space of dimension .
What is a Hermitian form on ?
Given a Hermitian form, define the matrix of the form with respect to the basis of , and describe in terms of the value of the Hermitian form on two elements of .
Now let be another basis of . Suppose , and let . Write down the matrix of the form with respect to this new basis in terms of and .
Let . Describe the dimension of in terms of the matrix .
(b) Write down the matrix of the real quadratic form
Using the Gram-Schmidt algorithm, find a basis which diagonalises the form. What are its rank and signature?
(c) Let be a real vector space, and , be the matrix of this form in some basis.
Prove that the signature of , minus the number of negative eigenvalues.
Explain, using an example, why the eigenvalues themselves depend on the choice of a basis.
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Paper 3 , Section I, H
2021 commentConsider a Markov chain on a state space .
(a) Define the notion of a communicating class. What does it mean for a communicating class to be closed?
(b) Taking , find the communicating classes associated with the transition matrix given by
and identify which are closed.
(c) Find the expected time for the Markov chain with transition matrix above to reach 6 starting from 1 .
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Paper 4, Section I, H
2021 commentShow that the simple symmetric random walk on is recurrent.
Three particles perform independent simple symmetric random walks on . What is the probability that they are all simultaneously at 0 infinitely often? Justify your answer.
[You may assume without proof that there exist constants such that for all positive integers
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Paper 1, Section II, 19H
2021 commentLet be a Markov chain with transition matrix . What is a stopping time of ? What is the strong Markov property?
The exciting game of 'Unopoly' is played by a single player on a board of 4 squares. The player starts with (where ). During each turn, the player tosses a fair coin and moves one or two places in a clockwise direction according to whether the coin lands heads or tails respectively. The player collects each time they pass (or land on) square 1. If the player lands on square 3 however, they immediately lose and go back to square 2. The game continues indefinitely unless the player is on square 2 with , in which case the player loses the game and the game ends.

(a) By setting up an appropriate Markov chain, show that if the player is at square 2 with , where , the probability that they are ever at square 2 with is
(b) Find the probability of losing the game when the player starts on square 1 with , where .
[Hint: Take the state space of your Markov chain to be .]
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Paper 2, Section II, 18H
2021 commentLet be a transition matrix on state space . What does it mean for a distribution to be an invariant distribution? What does it mean for and to be in detailed balance? Show that if and are in detailed balance, then is an invariant distribution.
(a) Assuming that an invariant distribution exists, state the relationship between this and
(i) the expected return time to a state ;
(ii) the expected time spent in a state between visits to a state .
(b) Let be a Markov chain with transition matrix where . The transition probabilities are given for by
where . For let . Compute the following, justifying your answers:
(i) The expected time spent in states between visits to state 1 ;
(ii) The expected time taken to return to state 1 , starting from 1 ;
(iii) The expected time taken to hit state 0 starting from
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Paper 2, Section I, C
2021 commentConsider the differential operator
acting on real functions with .
(i) Recast the eigenvalue equation in Sturm-Liouville form , identifying and .
(ii) If boundary conditions are imposed, show that the eigenvalues form an infinite discrete set and find the corresponding eigenfunctions for . If on is expanded in terms of your eigenfunctions i.e. , give an expression for . The expression can be given in terms of integrals that you need not evaluate.
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Paper 3, Section I, A
2021 commentLet be a -periodic function with Fourier expansion
Find the Fourier coefficients and for
Hence, or otherwise, find the Fourier coefficients and for the -periodic function defined by
Use your answers to evaluate
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Paper 1, Section II, C
2021 comment(a) By introducing the variables and (where is a constant), derive d'Alembert's solution of the initial value problem for the wave equation:
where and and are given functions (and subscripts denote partial derivatives).
(b) Consider the forced wave equation with homogeneous initial conditions:
where and is a given function. You may assume that the solution is given by
For the forced wave equation , now in the half space (and with as before), find (in terms of ) the solution for that satisfies the (inhomogeneous) initial conditions
and the boundary condition for .
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Paper 2, Section II, A
2021 commentThe Fourier transform of a function and its inverse are given by
(a) Calculate the Fourier transform of the function defined by:
(b) Show that the inverse Fourier transform of , for a positive real constant, is given by
(c) Consider the problem in the quarter plane :
Use the answers from parts (a) and (b) to show that
(d) Hence solve the problem in the quarter plane :
[You may quote without proof any property of Fourier transforms.]
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Paper 3, Section II, A
2021 commentLet be a solution of Legendre's equation with eigenvalue ,
such that and its derivatives , are regular at all points with .
(a) Show by induction that
for some constant . Find explicitly and show that its value is negative when is sufficiently large, for a fixed value of .
(b) Write the equation for in part (a) in self-adjoint form. Hence deduce that if is not identically zero, then .
[Hint: Establish a relation between integrals of the form and for certain functions and
(c) Use the results of parts (a) and (b) to show that if is a non-zero, regular solution of Legendre's equation on , then is a polynomial of degree and for some integer
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Paper 4, Section II, C
2021 commentThe function obeys the diffusion equation
Verify that
is a solution of , and by considering , find the solution having the initial form at .
Find, in terms of the error function, the solution of having the initial form
Sketch a graph of this solution at various times .
[The error function is
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Paper 1, Section I, B
2021 commentProve, from first principles, that there is an algorithm that can determine whether any real symmetric matrix is positive definite or not, with the computational cost (number of arithmetic operations) bounded by .
[Hint: Consider the LDL decomposition.]
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Paper 4, Section I, B
2021 comment(a) Given the data , find the interpolating cubic polynomial in the Newton form.
(b) We add to the data one more value, . Find the interpolating quartic polynomial for the extended data in the Newton form.
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Paper 1, Section II, B
2021 commentFor the ordinary differential equation
where and the function is analytic, consider an explicit one-step method described as the mapping
Here and with time step , producing numerical approximations to the exact solution of equation , with being the initial value of the numerical solution.
(i) Define the local error of a one-step method.
(ii) Let be a norm on and suppose that
for all , where is some positive constant. Let be given and denote the initial error (potentially non-zero). Show that if the local error of the one-step method ( ) is , then
(iii) Let and consider equation where is time-independent satisfying for all , where is a positive constant. Consider the one-step method given by
Use part (ii) to show that for this method we have that equation (††) holds (with a potentially different constant ) for .
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Paper 2, Section II, 17B
2021 comment(a) Define Householder reflections and show that a real Householder reflection is symmetric and orthogonal. Moreover, show that if , where is a Householder reflection and is a full matrix, then the computational cost (number of arithmetic operations) of computing can be operations, as opposed to for standard matrix products.
(b) Show that for any there exists an orthogonal matrix such that
In particular, has zero entries below the first subdiagonal. Show that one can compute such a and (they may not be unique) using arithmetic operations.
[Hint: Multiply A from the left and right with Householder reflections.]
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Paper 3, Section II, B
2021 commentThe functions are generated by the formula
(a) Show that is a monic polynomial of degree . Write down the explicit forms of .
(b) Demonstrate the orthogonality of these polynomials with respect to the scalar product
i.e. that for , and show that
where .
(c) Assuming that a three-term recurrence relation in the form
holds, find the explicit expressions for and as functions of .
[Hint: you may use the fact that
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Paper 1, Section I,
2021 comment(a) Let be a convex function for each . Show that
are both convex functions.
(b) Fix . Show that if is convex, then given by is convex.
(c) Fix vectors . Let be given by
Show that is convex. [You may use any result from the course provided you state it.]
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Paper 2, Section I, H
2021 commentFind the solution to the following Optimization problem using the simplex algorithm:
Write down the dual problem and give its solution.
-
Paper 3, Section II, H
2021 commentExplain what is meant by a two-person zero-sum game with payoff matrix , and define what is meant by an optimal strategy for each player. What are the relationships between the optimal strategies and the value of the game?
Suppose now that
Show that if strategy is optimal for player I, it must also be optimal for player II. What is the value of the game in this case? Justify your answer.
Explain why we must have for all . Hence or otherwise, find the optimal strategy and prove that it is unique.
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Paper 4, Section II, H
2021 comment(a) Consider the linear program
where and . What is meant by a basic feasible solution?
(b) Prove that if has a finite maximum, then there exists a solution that is a basic feasible solution.
(c) Now consider the optimization problem
subject to ,
where matrix and vectors are as in the problem , and . Suppose there exists a solution to . Further consider the linear program
(i) Suppose for all . Show that the maximum of is finite and at least as large as that of .
(ii) Suppose, in addition to the condition in part (i), that the entries of are strictly positive. Show that the maximum of is equal to that of .
(iii) Let be the set of basic feasible solutions of the linear program . Assuming the conditions in parts (i) and (ii) above, show that
[Hint: Argue that if is in the set of basic feasible solutions to , then
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Paper 3, Section I, C
2021 commentThe electron in a hydrogen-like atom moves in a spherically symmetric potential where is a positive constant and is the radial coordinate of spherical polar coordinates. The two lowest energy spherically symmetric normalised states of the electron are given by
where and is the mass of the electron. For any spherically symmetric function , the Laplacian is given by .
(i) Suppose that the electron is in the state and its energy is measured. Find the expectation value of the result.
(ii) Suppose now that the electron is in state (as above) at time . Let be the expectation value of a measurement of the electron's radial position at time . Show that the value of oscillates sinusoidally about a constant level and determine the frequency of the oscillation.
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Paper 4, Section I, C
2021 commentLet be the wavefunction for a particle of mass moving in one dimension in a potential . Show that, with suitable boundary conditions as ,
Why is this important for the interpretation of quantum mechanics?
Verify the result above by first calculating for the free particle solution
where and are real constants, and then considering the resulting integral.
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Paper 1, Section II, C
2021 commentConsider a quantum mechanical particle of mass in a one-dimensional stepped potential well given by:
where and are constants.
(i) Show that all energy levels of the particle are non-negative. Show that any level with satisfies
where
(ii) Suppose that initially and the particle is in the ground state of the potential well. is then changed to a value (while the particle's wavefunction stays the same) and the energy of the particle is measured. For , give an expression in terms of for prob , the probability that the energy measurement will find the particle having energy . The expression may be left in terms of integrals that you need not evaluate.
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Paper 2, Section II, C
2021 comment(a) Write down the expressions for the probability density and associated current density of a quantum particle in one dimension with wavefunction . Show that if is a stationary state then the function is constant.
For the non-normalisable free particle wavefunction (where and are real constants and is a complex constant) compute the functions and , and briefly give a physical interpretation of the functions and in this case.
(b) A quantum particle of mass and energy moving in one dimension is incident from the left in the potential given by
where and are positive constants. Write down the form of the wavefunction in the regions and .
Suppose now that . Show that the probability of transmission of the particle into the region is given by
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Paper 4, Section II, C
2021 comment(a) Consider the angular momentum operators and where
Use the standard commutation relations for these operators to show that
Deduce that if is a joint eigenstate of and with angular momentum quantum numbers and respectively, then are also joint eigenstates, provided they are non-zero, with quantum numbers and .
(b) A harmonic oscillator of mass in three dimensions has Hamiltonian
Find eigenstates of in terms of eigenstates for an oscillator in one dimension with and eigenvalues ; hence determine the eigenvalues of .
Verify that the ground state for is a joint eigenstate of and with . At the first excited energy level, find an eigenstate of with and construct from this two eigenstates of with .
Why should you expect to find joint eigenstates of and ?
[ The first two eigenstates for an oscillator in one dimension are and , where and are normalisation constants. ]
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Paper 1, Section I, H
2021 commentLet be i.i.d. Bernoulli random variables, where and is unknown.
(a) What does it mean for a statistic to be sufficient for ? Find such a sufficient statistic .
(b) State and prove the Rao-Blackwell theorem.
(c) By considering the estimator of , find an unbiased estimator of that is a function of the statistic found in part (a), and has variance strictly smaller than that of .
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Paper 2, Section I,
2021 commentThe efficacy of a new drug was tested as follows. Fifty patients were given the drug, and another fifty patients were given a placebo. A week later, the numbers of patients whose symptoms had gone entirely, improved, stayed the same and got worse were recorded, as summarised in the following table.
\begin{tabular}{|c|c|c|} \hline & Drug & Placebo \ \hline symptoms gone & 14 & 6 \ improved & 21 & 19 \ same & 10 & 10 \ worse & 5 & 15 \ \hline \end{tabular}
Conduct a significance level test of the null hypothesis that the medicine and placebo have the same effect, against the alternative that their effects differ.
[Hint: You may find some of the following values relevant:
\begin{tabular}{|c|cccccc|} \hline Distribution & & & & & & \ \hline 95 th percentile & & & & & & \ \hline \end{tabular}
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Paper 1, Section II, H
2021 comment(a) Show that if are independent random variables with common distribution, then . [Hint: If then if and otherwise.]
(b) Show that if then .
(c) State the Neyman-Pearson lemma.
(d) Let be independent random variables with common density proportional to for . Find a most powerful test of size of against , giving the critical region in terms of a quantile of an appropriate gamma distribution. Find a uniformly most powerful test of size of against .
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Paper 1, Section II, I
2021 commentLet be an algebraically closed field and let be a non-empty affine variety. Show that is a finite union of irreducible subvarieties.
Let and be subvarieties of given by the vanishing loci of ideals and respectively. Prove the following assertions.
(i) The variety is equal to the vanishing locus of the ideal .
(ii) The variety is equal to the vanishing locus of the ideal .
Decompose the vanishing locus
into irreducible components.
Let be the union of the three coordinate axes. Let be the union of three distinct lines through the point in . Prove that is not isomorphic to .
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Paper 2, Section II, I
2021 commentLet be an algebraically closed field and . Exhibit as an open subset of affine space . Deduce that is smooth. Prove that it is also irreducible.
Prove that is isomorphic to a closed subvariety in an affine space.
Show that the matrix multiplication map
that sends a pair of matrices to their product is a morphism.
Prove that any morphism from to is constant.
Prove that for any morphism from to is constant.
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Paper 3, Section II, I
2021 commentIn this question, all varieties are over an algebraically closed field of characteristic zero.
What does it mean for a projective variety to be smooth? Give an example of a smooth affine variety whose projective closure is not smooth.
What is the genus of a smooth projective curve? Let be the hypersurface . Prove that contains a smooth curve of genus
Let be an irreducible curve of degree 2 . Prove that is isomorphic to .
We define a generalized conic in to be the vanishing locus of a non-zero homogeneous quadratic polynomial in 3 variables. Show that there is a bijection between the set of generalized conics in and the projective space , which maps the conic to the point whose coordinates are the coefficients of .
(i) Let be the subset of conics that consist of unions of two distinct lines. Prove that is not Zariski closed, and calculate its dimension.
(ii) Let be the homogeneous ideal of polynomials vanishing on . Determine generators for the ideal .
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Paper 4, Section II, I
2021 commentLet be a smooth irreducible projective algebraic curve over an algebraically closed field.
Let be an effective divisor on . Prove that the vector space of rational functions with poles bounded by is finite dimensional.
Let and be linearly equivalent divisors on . Exhibit an isomorphism between the vector spaces and .
What is a canonical divisor on ? State the Riemann-Roch theorem and use it to calculate the degree of a canonical divisor in terms of the genus of .
Prove that the canonical divisor on a smooth cubic plane curve is linearly equivalent to the zero divisor.
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Paper 1, Section II, 21F
2021 comment(a) What does it mean for two spaces and to be homotopy equivalent?
(b) What does it mean for a subspace to be a retract of a space ? What does it mean for a space to be contractible? Show that a retract of a contractible space is contractible.
(c) Let be a space and a subspace. We say the pair has the homotopy extension property if, for any pair of maps and with
there exists a map with
Now suppose that is contractible. Denote by the quotient of by the equivalence relation if and only if or . Show that, if satisfies the homotopy extension property, then and are homotopy equivalent.
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Paper 2, Section II, 21F
2021 comment(a) State a suitable version of the Seifert-van Kampen theorem and use it to calculate the fundamental groups of the torus and of the real projective plane .
(b) Show that there are no covering maps or .
(c) Consider the following covering space of :

Here the line segments labelled and are mapped to the two different copies of contained in , with orientations as indicated.
Using the Galois correspondence with basepoints, identify a subgroup of
(where is the wedge point) that corresponds to this covering space.
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Paper 3, Section II, 20F
2021 commentLet be a space. We define the cone of to be
where if and only if either or .
(a) Show that if is triangulable, so is . Calculate . [You may use any results proved in the course.]
(b) Let be a simplicial complex and a subcomplex. Let , and let be the space obtained by identifying with . Show that there is a long exact sequence
(c) In part (b), suppose that and for some . Calculate for all .
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Paper 4, Section II, 21F
2021 comment(a) Define the Euler characteristic of a triangulable space .
(b) Let be an orientable surface of genus . A is a doublebranched cover if there is a set of branch points, such that the restriction is a covering map of degree 2 , but for each , consists of one point. By carefully choosing a triangulation of , use the Euler characteristic to find a formula relating and .
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Paper 1, Section II,
2021 commentBelow, is the -algebra of Lebesgue measurable sets and is Lebesgue measure.
(a) State the Lebesgue differentiation theorem for an integrable function . Let be integrable and define by for some . Show that is differentiable -almost everywhere.
(b) Suppose is strictly increasing, continuous, and maps sets of -measure zero to sets of -measure zero. Show that we can define a measure on by setting for , and establish that . Deduce that is differentiable -almost everywhere. Does the result continue to hold if is assumed to be non-decreasing rather than strictly increasing?
[You may assume without proof that a strictly increasing, continuous, function is injective, and is continuous.]
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Paper 2, Section II, H
2021 commentDefine the Schwartz space, , and the space of tempered distributions, , stating what it means for a sequence to converge in each space.
For a function , and non-negative integers , we say if
You may assume that equipped with is a Banach space in which is dense.
(a) Show that if there exist and such that
Deduce that there exists a unique such that for all .
(b) Recall that is positive if for all satisfying . Show that if is positive, then there exist and such that
Hint: Note that
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Paper 3, Section II, H
2021 comment(a) State the Riemann-Lebesgue lemma. Show that the Fourier transform maps to itself continuously.
(b) For some , let . Consider the following system of equations for
Show that there exists a unique solving the equations with for . You need not find explicitly, but should give an expression for the Fourier transform of . Show that there exists a constant such that
For what values of can we conclude that ?
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Paper 4, Section II,
2021 commentFix and let satisfy
(a) Let be a sequence of functions in . For , what is meant by (i) in and (ii) in ? Show that if , then
(b) Suppose that is a sequence with , and that there exists such that for all . Show that there exists and a subsequence , such that for any sequence with and , we have
Give an example to show that the result need not hold if the condition is replaced by in .
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Paper 1, Section II, B
2021 comment(a) Discuss the variational principle that allows one to derive an upper bound on the energy of the ground state for a particle in one dimension subject to a potential .
If , how could you adapt the variational principle to derive an upper bound on the energy of the first excited state?
(b) Consider a particle of mass (in certain units) subject to a potential
(i) Using the trial wavefunction
with , derive the upper bound , where
(ii) Find the zero of in and show that any extremum must obey
(iii) By sketching or otherwise, deduce that there must always be a minimum in . Hence deduce the existence of a bound state.
(iv) Working perturbatively in , show that
[Hint: You may use that for
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Paper 2, Section II, 36B
2021 comment(a) The -wave solution for the scattering problem of a particle of mass and momentum has the asymptotic form
Define the phase shift and verify that .
(b) Define the scattering amplitude . For a spherically symmetric potential of finite range, starting from , derive the expression
giving the cross-section in terms of the phase shifts of the partial waves.
(c) For with , show that a bound state exists and compute its energy. Neglecting the contributions from partial waves with , show that
(d) For with compute the -wave contribution to . Working to leading order in , show that has a local maximum at . Interpret this fact in terms of a resonance and compute its energy and decay width.
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Paper 3, Section II, 34B
2021 comment(a) In three dimensions, define a Bravais lattice and its reciprocal lattice .
A particle is subject to a potential with for and . State and prove Bloch's theorem and specify how the Brillouin zone is related to the reciprocal lattice.
(b) A body-centred cubic lattice consists of the union of the points of a cubic lattice and all the points at the centre of each cube:
where and are unit vectors parallel to the Cartesian coordinates in . Show that is a Bravais lattice and determine the primitive vectors and .
Find the reciprocal lattice Briefly explain what sort of lattice it is.
Hint: The matrix has inverse .
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Paper 4, Section II, B
2021 comment(a) Consider the nearly free electron model in one dimension with mass and periodic potential with and
Ignoring degeneracies, the energy spectrum of Bloch states with wavenumber is
where are normalized eigenstates of the free Hamiltonian with wavenumber . What is in this formula?
If we impose periodic boundary conditions on the wavefunctions, with and a positive integer, what are the allowed values of and ? Determine for these allowed values.
(b) State when the above expression for ceases to be a good approximation and explain why. Quoting any result you need from degenerate perturbation theory, calculate to the location and width of the band gaps.
(c) Determine the allowed energy bands for each of the potentials
(d) Briefly discuss a macroscopic physical consequence of the existence of energy bands.
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Paper 1, Section II, 28K
2021 commentThe particles of an Ideal Gas form a spatial Poisson process on with constant intensity , called the activity of the gas.
(a) Prove that the independent mixture of two Ideal Gases with activities and is again an Ideal Gas. What is its activity? [You must prove any results about Poisson processes that you use. The independent mixture of two gases with particles and is given by
(b) For an Ideal Gas of activity , find the limiting distribution of
as for a given sequence of subsets with .
(c) Let be a smooth non-negative function vanishing outside a bounded subset of . Find the mean and variance of , where the sum runs over the particles of an ideal gas of activity . [You may use the properties of spatial Poisson processes established in the lectures.]
[Hint: recall that the characteristic function of a Poisson random variable with mean is
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Paper 2, Section II,
2021 commentLet be an irreducible, non-explosive, continuous-time Markov process on the state space with generator .
(a) Define its jump chain and prove that it is a discrete-time Markov chain.
(b) Define what it means for to be recurrent and prove that is recurrent if and only if its jump chain is recurrent. Prove also that this is the case if the transition semigroup satisfies
(c) Show that is recurrent for at least one of the following generators:
[Hint: You may use that the semigroup associated with a -matrix on such that depends only on (and has sufficient decay) can be written as
where . You may also find the bound useful.
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Paper 3, Section II,
2021 comment(a) Customers arrive at a queue at the event times of a Poisson process of rate . The queue is served by two independent servers with exponential service times with parameter each. If the queue has length , an arriving customer joins with probability and leaves otherwise (where . For which and is there a stationary distribution?
(b) A supermarket allows a maximum of customers to shop at the same time. Customers arrive at the event times of a Poisson process of rate 1 , they enter the supermarket when possible, and they leave forever for another supermarket otherwise. Customers already in the supermarket pay and leave at the event times of an independent Poisson process of rate . When is there a unique stationary distribution for the number of customers in the supermarket? If it exists, find it.
(c) In the situation of part (b), started from equilibrium, show that the departure process is Poissonian.
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Paper 4, Section II,
2021 commentLet be a continuous-time Markov process with state space and generator satisfying for all . The local time up to time of is the random vector defined by
(a) Let be any function that is differentiable with respect to its second argument, and set
Show that
where
(b) For , write for the vector of squares of the components of . Let be a function such that whenever for some fixed . Using integration by parts, or otherwise, show that for all
where denotes .
(c) Let be a function with whenever for some fixed . Given , now let
in part (b) and deduce, using part (a), that
[You may exchange the order of integrals and derivatives without justification.]
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Paper 2, Section II, 32A
2021 comment(a) Let and , for , be real-valued functions on .
(i) Define what it means for the sequence to be an asymptotic sequence as .
(ii) Define what it means for to have the asymptotic expansion
(b) Use the method of stationary phase to calculate the leading-order asymptotic approximation as of
[You may assume that .]
(c) Use Laplace's method to calculate the leading-order asymptotic approximation as of
[In parts (b) and (c) you should include brief qualitative reasons for the origin of the leading-order contributions, but you do not need to give a formal justification.]
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Paper 3, Section II, 30A
2021 comment(a) Carefully state Watson's lemma.
(b) Use the method of steepest descent and Watson's lemma to obtain an infinite asymptotic expansion of the function
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Paper 4, Section II, A
2021 comment(a) Classify the nature of the point at for the ordinary differential equation
(b) Find a transformation from to an equation of the form
and determine .
(c) Given satisfies ( , use the Liouville-Green method to find the first three terms in an asymptotic approximation as for , verifying the consistency of any approximations made.
(d) Hence obtain corresponding asymptotic approximations as of two linearly independent solutions of .
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Paper 1, Section I, F
2021 commentLet be the partial function on variables that is computed by the th machine (or the empty function if does not encode a machine).
Define the halting set .
Given , what is a many-one reduction of to ?
State the theorem and use it to show that a subset of is recursively enumerable if and only if .
Give an example of a set with but .
[You may assume that is recursively enumerable and that .]
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Paper 2, Section I, F
2021 commentAssuming the definition of a deterministic finite-state automaton (DFA) , what is the extended transition function for ? Also assuming the definition of a nondeterministic finite-state automaton (NFA) , what is in this case?
Define the languages accepted by and , respectively, in terms of .
Given an NFA as above, describe the subset construction and show that the resulting DFA accepts the same language as . If has one accept state then how many does have?
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Paper 3, Section I, F
2021 commentDefine a regular expression and explain how this gives rise to a language .
Define a deterministic finite-state automaton and the language that it accepts.
State the relationship between languages obtained from regular expressions and languages accepted by deterministic finite-state automata.
Let and be regular languages. Is always regular? What about ?
Now suppose that are regular languages. Is the countable union always regular? What about the countable intersection ?
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Paper 4, Section I,
2021 commentState the pumping lemma for regular languages.
Which of the following languages over the alphabet are regular?
(i) .
(ii) where is the reverse of the word .
(iii) does not contain the subwords 01 or 10.
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Paper 1, Section II, F
2021 commentFor give the definition of a partial recursive function in terms of basic functions, composition, recursion and minimisation.
Show that the following partial functions from to are partial recursive: (i) (ii) (iii)
Which of these can be defined without using minimisation?
What is the class of functions that can be defined using only basic functions and composition? [Hint: See which functions you can obtain and then show that these form a class that is closed with respect to the above.]
Show directly that every function in this class is computable.
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Paper 3, Section II,
2021 commentSuppose that is a context-free grammar without -productions. Given a derivation of some word in the language of , describe a parse tree for this derivation.
State and prove the pumping lemma for . How would your proof differ if you did not assume that was in Chomsky normal form, but merely that has no - or unit productions?
For the alphabet of terminal symbols, state whether the following languages over are context free, giving reasons for your answer. (i) , (ii) , (iii) .
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Paper 1, Section I, D
2021 commentTwo equal masses move along a straight line between two stationary walls. The mass on the left is connected to the wall on its left by a spring of spring constant , and the mass on the right is connected to the wall on its right by a spring of spring constant . The two masses are connected by a third spring of spring constant .
(a) Show that the Lagrangian of the system can be written in the form
where , for , are the displacements of the two masses from their equilibrium positions, and and are symmetric matrices that should be determined.
(b) Let
where and . Using Lagrange's equations of motion, show that the angular frequencies of the normal modes of the system are given by
where
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Paper 2, Section I, D
2021 commentShow that, in a uniform gravitational field, the net gravitational torque on a system of particles, about its centre of mass, is zero.
Let be an inertial frame of reference, and let be the frame of reference with the same origin and rotating with angular velocity with respect to . You may assume that the rates of change of a vector observed in the two frames are related by
Derive Euler's equations for the torque-free motion of a rigid body.
Show that the general torque-free motion of a symmetric top involves precession of the angular-velocity vector about the symmetry axis of the body. Determine how the direction and rate of precession depend on the moments of inertia of the body and its angular velocity.
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Paper 3 , Section I, D
2021 commentThe Lagrangian of a particle of mass and charge in an electromagnetic field takes the form
Explain the meaning of and , and how they are related to the electric and magnetic fields.
Obtain the canonical momentum and the Hamiltonian .
Suppose that the electric and magnetic fields have Cartesian components and , respectively, where and are positive constants. Explain why the Hamiltonian of the particle can be taken to be
State three independent integrals of motion in this case.
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Paper 4, Section I, D
2021 commentBriefly describe a physical object (a Lagrange top) whose Lagrangian is
Explain the meaning of the symbols in this equation.
Write down three independent integrals of motion for this system, and show that the nutation of the top is governed by the equation
where and is a certain cubic function that you need not determine.
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Paper 2, Section II, D
2021 comment(a) Show that the Hamiltonian
where is a positive constant, describes a simple harmonic oscillator with angular frequency . Show that the energy and the action of the oscillator are related by .
(b) Let be a constant. Verify that the differential equation
is solved by
when , where is a constant you should determine in terms of .
(c) Show that the solution in part (b) obeys
Hence show that the fractional variation of the action in the limit is , but that these variations do not accumulate. Comment on this behaviour in relation to the theory of adiabatic invariance.
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Paper 4, Section II, 15D
2021 comment(a) Let be a set of canonical phase-space variables for a Hamiltonian system with degrees of freedom. Define the Poisson bracket of two functions and . Write down the canonical commutation relations that imply that a second set of phase-space variables is also canonical.
(b) Consider the near-identity transformation
where and are small. Determine the approximate forms of the canonical commutation relations, accurate to first order in and . Show that these are satisfied when
where is a small parameter and is some function of the phase-space variables.
(c) In the limit this near-identity transformation is called the infinitesimal canonical transformation generated by . Let be an autonomous Hamiltonian. Show that the change in the Hamiltonian induced by the infinitesimal canonical transformation is
Explain why is an integral of motion if and only if the Hamiltonian is invariant under the infinitesimal canonical transformation generated by .
(d) The Hamiltonian of the gravitational -body problem in three-dimensional space is
where and are the mass, position and momentum of body . Determine the form of and the infinitesimal canonical transformation that correspond to the translational symmetry of the system.
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Paper 1, Section I,
2021 commentLet be an code. Define the parameters and . In each of the following cases define the new code and give its parameters.
(i) is the parity extension of .
(ii) is the punctured code (assume ).
(iii) is the shortened code (assume ).
Let . Suppose the parity extension of is transmitted through a binary symmetric channel where is the probability of a single-bit error in the channel. Calculate the probability that an error in the transmission of a single codeword is not noticed.
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Paper 2, Section I, K
2021 commentState Shannon's noisy coding theorem for a binary symmetric channel, defining the terms involved.
Suppose a channel matrix, with output alphabet of size , is such that the entries in each row are the elements of the set in some order. Further suppose that all columns are permutations of one another. Show that the channel's information capacity is given by
Show that the information capacity of the channel matrix
is given by .
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Paper 3, Section I, K
2021 commentLet . Define the Hamming code of length . Explain what it means to be a perfect code and show that is a perfect code.
Suppose you are using the Hamming code of length and you receive the message of length . How would you decode this message using minimum distance decoding? Explain why this leads to correct decoding if at most one channel error has occurred.
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Paper 4 , Section I,
2021 commentDescribe the Rabin scheme for coding a message as modulo a certain integer .
Describe the RSA encryption scheme with public key and private key .
[In both cases you should explain how you encrypt and decrypt.]
Give an advantage and a disadvantage that the Rabin scheme has over the RSA scheme.
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Paper 1, Section II,
2021 commentLet be a finite alphabet and a random variable that takes each value with probability . Define the entropy of .
Suppose and is a decipherable code. Write down an expression for the expected word length of .
Prove that the minimum expected word length of a decipherable code satisfies
[You can use Kraft's and Gibbs' inequalities as long as they are clearly stated.]
Suppose a decipherable binary code has word lengths . Show that
Suppose is a source that emits sourcewords and is the probability that is emitted, where . Let and for . Let for . Now define a code by where is the (fractional part of the) binary expansion of to decimal places. Prove that this defines a decipherable code.
What does it mean for a code to be optimal? Is the code defined in the previous paragraph in terms of the necessarily optimal? Justify your answer.
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Paper 2, Section II,
2021 comment(a) Define what it means to say that is a binary cyclic code. Explain the bijection between the set of binary cyclic codes of length and the factors of in .
(b) What is a linear feedback shift register?
Suppose that is a linear feedback shift register. Further suppose and is a positive integer such that . Let be the matrix . Considering as a parity check matrix of a code , show that is a binary cyclic code.
(c) Suppose that is a binary cyclic code. Prove that, if does not contain the codeword , then all codewords in have even weight.
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Paper 1, Section I, 9B
2021 commentThe continuity, Euler and Poisson equations governing how non-relativistic fluids with energy density , pressure and velocity propagate in an expanding universe take the form
where and is the scale factor.
(a) Show that, for a homogeneous and isotropic flow with and , consistency of the Euler equation with the Poisson equation implies Raychaudhuri's equation.
(b) Explain why this derivation of Raychaudhuri's equation is an improvement over the derivation of the Friedmann equation using only Newtonian gravity.
(c) Consider small perturbations about a homogeneous and isotropic flow,
with . Show that, to first order in , the continuity equation can be written as
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Paper 2, Section I, 9B
2021 comment(a) The generalised Boltzmann distribution is given by
where is the chemical potential,
Find the average particle number with momentum , assuming that all particles have rest mass and are either
(i) bosons, or
(ii) fermions .
(b) The photon total number density is given by
where . Consider now the fractional ionisation of hydrogen
In our universe , where is the baryon-to-photon number density. Find an expression for the ratio
in terms of , the electron mass , the speed of light and the ionisation energy of hydrogen .
One might expect neutral hydrogen to form at a temperature , but instead in our universe it happens at the much lower temperature . Briefly explain why this happens.
[You may use without proof the Saha equation
for chemical equilibrium in the reaction
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Paper 3, Section I, 9B
2021 commentThe expansion of the universe during inflation is governed by the Friedmann equation
and the equation of motion for the inflaton field ,
Consider the potential
with and .
(a) Show that the inflationary equations have the exact solution
for arbitrary and appropriate choices of and . Determine the range of for which the solution exists. For what values of does inflation occur?
(b) Using the inflaton equation of motion and
together with the continuity equation
determine .
(c) What is the range of the pressure energy density ratio for which inflation occurs?
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Paper 4, Section I, B
2021 commentA collection of particles, with masses and positions , interact through a gravitational potential
Assume that the system is gravitationally bound, and that the positions and velocities are bounded for all time. Further, define the time average of a quantity by
(a) Assuming that the time average of the kinetic energy and potential energy are well defined, show that
[You should consider the quantity , with all measured relative to the centre of mass.]
(b) Explain how part (a) can be used, together with observations, to provide evidence in favour of dark matter. [You may assume that time averaging may be replaced by an average over particles.]
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Paper 1, Section II, 15B
2021 comment(a) Consider the following action for the inflaton field
Use the principle of least action to derive the equation of motion for the inflaton ,
where . [In the derivation you may discard boundary terms.]
(b) Consider a regime where is approximately constant so that the universe undergoes a period of exponential expansion during which . Show that can be written in terms of the spatial Fourier transform of as
(c) Define conformal time and determine the range of when . Show that can be written in terms of the conformal time as
(d) Let denote the state that in the far past was in the ground state of the standard harmonic oscillator with frequency . Assuming that the quantum variance of is given by
explain in which sense inflation naturally generates a scale-invariant power spectrum. [You may use that has dimensions of [length
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Paper 3, Section II, B
2021 comment(a) Consider a closed universe endowed with cosmological constant and filled with radiation with pressure and energy density . Using the equation of state and the continuity equation
determine how depends on . Give the physical interpretation of the scaling of with
(b) For such a universe the Friedmann equation reads
What is the physical meaning of
(c) Making the substitution , determine and such that the Friedmann equation takes the form
Using the substitution and the boundary condition , deduce the boundary condition for .
Show that
and hence that
Express the constant in terms of and .
Sketch the graphs of for the cases and .
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Paper 1, Section II, 26F
2021 comment(a) Let be a surface. Give a parametrisation-free definition of the first fundamental form of . Use this definition to derive a description of it in terms of the partial derivatives of a local parametrisation .
(b) Let be a positive constant. Show that the half-cone
is locally isometric to the Euclidean plane. [Hint: Use polar coordinates on the plane.]
(c) Define the second fundamental form and the Gaussian curvature of . State Gauss' Theorema Egregium. Consider the set
(i) Show that is a surface.
(ii) Calculate the Gaussian curvature of at each point. [Hint: Complete the square.]
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Paper 2, Section II,
2021 commentLet be a domain in , and let be a smooth map. Define what it means for to be an immersion. What does it mean for an immersion to be isothermal?
Write down a formula for the mean curvature of an immersion in terms of the first and second fundamental forms. What does it mean for an immersed surface to be minimal? Assume that is an isothermal immersion. Prove that it is minimal if and only if are harmonic functions of .
For , and smooth functions , assume that
is an isothermal immersion. Find all possible pairs such that this immersion is minimal.
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Paper 3, Section II, F
2021 commentLet and be smooth boundaryless manifolds. Suppose is a smooth map. What does it mean for to be a regular value of ? State Sard's theorem and the stack-of-records theorem.
Suppose is another smooth map. What does it mean for and to be smoothly homotopic? Assume now that is compact, and has the same dimension as . Suppose that is a regular value for both and . Prove that
Let be a non-empty open subset of the sphere. Suppose that is a smooth map such that for all . Show that there must exist a pair of antipodal points on which is mapped to another pair of antipodal points by .
[You may assume results about compact 1-manifolds provided they are accurately stated.]
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Paper 4, Section II, F
2021 commentLet be an interval, and be a surface. Assume that is a regular curve parametrised by arc-length. Define the geodesic curvature of . What does it mean for to be a geodesic curve?
State the global Gauss-Bonnet theorem including boundary terms.
Suppose that is a surface diffeomorphic to a cylinder. How large can the number of simple closed geodesics on be in each of the following cases?
(i) has Gaussian curvature everywhere zero;
(ii) has Gaussian curvature everywhere positive;
(iii) has Gaussian curvature everywhere negative.
In cases where there can be two or more simple closed geodesics, must they always be disjoint? Justify your answer.
[A formula for the Gaussian curvature of a surface of revolution may be used without proof if clearly stated. You may also use the fact that a piecewise smooth curve on a cylinder without self-intersections either bounds a domain homeomorphic to a disc or is homotopic to the waist-curve of the cylinder.]
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Paper 1, Section II, A
2021 comment(a) State the properties defining a Lyapunov function for a dynamical system . State Lyapunov's first theorem and La Salle's invariance principle.
(b) Consider the system
Show that for the origin is asymptotically stable, stating clearly any arguments that you use.
(c) Sketch the phase plane, (i) for and (ii) for , giving brief details of any reasoning and identifying the fixed points. Include the domain of stability of the origin in your sketch for case (ii).
(d) For show that the trajectory with , where , satisfies for . Show also that, for any , the trajectory cannot remain outside the region .
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Paper 2, Section II, A
2021 commentConsider a modified van der Pol system defined by
where and are constants.
(a) A parallelogram PQRS of width is defined by
where . Show that if is sufficiently large then trajectories never leave the region inside the parallelogram.
Hence show that if there must be a periodic orbit. Explain your reasoning carefully.
(b) Use the energy-balance method to analyse the behaviour of the system for , identifying the difference in behaviours between and .
(c) Describe the behaviour of the system for , using sketches of the phase plane to illustrate your arguments for the cases and .
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Paper 3, Section II, A
2021 commentConsider the system
where and are constants with .
(a) Find the fixed points, and classify those on . State how the number of fixed points depends on and . Hence, or otherwise, deduce the values of at which stationary bifurcations occur for fixed .
(b) Sketch bifurcation diagrams in the -plane for the cases and , indicating the stability of the fixed points and the type of the bifurcations in each case. [You are not required to prove that the stabilities or bifurcation types are as you indicate.]
(c) For the case , analyse the bifurcation at using extended centre manifold theory and verify that the evolution equation on the centre manifold matches the behaviour you deduced from the bifurcation diagram in part (b).
(d) For , sketch the phase plane in the immediate neighbourhood of where the bifurcation of part (c) occurs.
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Paper 4, Section II, A
2021 comment(a) A continuous map of an interval into itself has a periodic orbit of period 3 . Prove that also has periodic orbits of period for all positive integers .
(b) What is the minimum number of distinct orbits of of periods 2,4 and 5 ? Explain your reasoning with a directed graph. [Formal proof is not required.]
(c) Consider the piecewise linear map defined by linear segments between and . Calculate the orbits of periods 2,4 and 5 that are obtained from the directed graph in part (b).
[In part (a) you may assume without proof:
(i) If and are non-empty closed bounded intervals such that then there is a closed bounded interval such that .
(ii) The Intermediate Value Theorem.]
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Paper 1, Section II, 37C
2021 comment(a) An electromagnetic field is specified by a four-vector potential
Define the corresponding field-strength tensor and state its transformation property under a general Lorentz transformation.
(b) Write down two independent Lorentz scalars that are quadratic in the field strength and express them in terms of the electric and magnetic fields, and . Show that both these scalars vanish when evaluated on an electromagnetic plane-wave solution of Maxwell's equations of arbitrary wavevector and polarisation.
(c) Find (non-zero) constant, homogeneous background fields and such that both the Lorentz scalars vanish. Show that, for any such background, the field-strength tensor obeys
(d) Hence find the trajectory of a relativistic particle of mass and charge in this background. You should work in an inertial frame where the particle is at rest at the origin at and in which .
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Paper 3, Section II, 36C
2021 comment(a) Derive the Larmor formula for the total power emitted through a large sphere of radius by a non-relativistic particle of mass and charge with trajectory . You may assume that the electric and magnetic fields describing radiation due to a source localised near the origin with electric dipole moment can be approximated as
Here, the radial distance is assumed to be much larger than the wavelength of emitted radiation which, in turn, is large compared to the spatial extent of the source.
(b) A non-relativistic particle of mass , moving at speed along the -axis in the positive direction, encounters a step potential of width and height described by
where is a monotonically increasing function with and . The particle carries charge and loses energy by emitting electromagnetic radiation. Assume that the total energy loss through emission is negligible compared with the particle's initial kinetic energy . For , show that the total energy lost is
Find the total energy lost also for the case .
(c) Take and explicitly evaluate the particle energy loss in each of the cases and . What is the maximum value attained by as is varied?
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Paper 4, Section II, 36C
2021 comment(a) Define the electric displacement for a medium which exhibits a linear response with polarisation constant to an applied electric field with polarisation constant . Write down the effective Maxwell equation obeyed by in the timeindependent case and in the absence of any additional mobile charges in the medium. Describe appropriate boundary conditions for the electric field at an interface between two regions with differing values of the polarisation constant. [You should discuss separately the components of the field normal to and tangential to the interface.]
(b) Consider a sphere of radius , centred at the origin, composed of dielectric material with polarisation constant placed in a vacuum and subjected to a constant, asymptotically homogeneous, electric field, with as . Using the ansatz
with constants and to be determined, find a solution to Maxwell's equations with appropriate boundary conditions at .
(c) By comparing your solution with the long-range electric field due to a dipole consisting of electric charges located at displacements find the induced electric dipole moment of the dielectric sphere.
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Paper 1, Section II, 39A II
2021 comment(a) Write down the Stokes equations for the motion of an incompressible viscous fluid with negligible inertia (in the absence of body forces). What does it mean that Stokes flow is linear and reversible?
(b) The region between two concentric rigid spheres of radii and is filled with fluid of large viscosity . The outer sphere is held stationary, while the inner sphere is made to rotate with angular velocity .
(i) Use symmetry and the properties of Stokes flow to deduce that , where is the pressure due to the flow.
(ii) Verify that both solid-body rotation and satisfy the Stokes equations with . Hence determine the fluid velocity between the spheres.
(iii) Calculate the stress tensor in the flow.
(iv) Deduce that the couple exerted by the fluid in on the fluid in , where , is given by
independent of the value of . [Hint: Do not substitute the form of and in until the end of the calculation.]
Comment on the form of this result for and for .
may use , where is the normal to
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Paper 2, Section II, 39A II
2021 comment(a) Incompressible fluid of viscosity fills the thin, slowly varying gap between rigid boundaries at and . The boundary at translates in its own plane with a constant velocity , while the other boundary is stationary. If has typical magnitude and varies on a lengthscale , state conditions for the lubrication approximation to be appropriate.
Write down the lubrication equations for this problem and show that the horizontal volume flux is given by
where is the pressure.
Explain why for some function . Deduce that satisfies the equation
(b) Now consider the case for and for , where and are constants, and are polar coordinates. A uniform pressure gradient is applied at infinity. Show that as , where the constant is to be determined.
Given that , you may assume that the equations of part (a) apply for and , and are subject to conditions that the radial component of the volume flux and the pressure are both continuous across . Show that these continuity conditions imply that
respectively, where []denotes the jump across .
Hence determine and deduce that the total flux through is given by
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Paper 3, Section II, 38A II
2021 commentViscous fluid occupying is bounded by a rigid plane at and is extracted through a small hole at the origin at a constant flow rate . Assume that for sufficiently small values of the velocity is well-approximated by
except within a thin axisymmetric boundary layer near .
(a) Estimate the Reynolds number of the flow as a function of , and thus give an estimate for how small needs to be for such a solution to be applicable. Show that the radial pressure gradient is proportional to .
(b) In cylindrical polar coordinates , the steady axisymmetric boundary-layer equations for the velocity components can be written as
and is the Stokes streamfunction. Verify that the condition of incompressibility is satisfied by the use of .
Use scaling arguments to estimate the thickness of the boundary layer near and then to motivate seeking a similarity solution of the form
(c) Obtain the differential equation satisfied by , and state the conditions that would determine its solution. [You are not required to find this solution.]
By considering the flux in the boundary layer, explain why there should be a correction to the approximation of relative magnitude .
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Paper 4, Section II, A II
2021 commentConsider a steady axisymmetric flow with components in cylindrical polar coordinates , where is a positive constant. The fluid has density and kinematic viscosity .
(a) Briefly describe the flow and confirm that it is incompressible.
(b) Show that the vorticity has one component , in the direction. Write down the corresponding vorticity equation and derive the solution
Hence find and show that it has a maximum at some finite radius , indicating how scales with and .
(c) Find an expression for the net advection of angular momentum, prv, into the finite cylinder defined by and . Show that this is always positive and asymptotes to the value
as
(d) Show that the torque exerted on the cylinder of part (c) by the exterior flow is always negative and demonstrate that it exactly balances the net advection of angular momentum. Comment on why this has to be so.
[You may assume that for a flow in cylindrical polar coordinates
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Paper 1, Section I, 7E
2021 commentEvaluate the integral
stating clearly any standard results involving contour integrals that you use.
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Paper 2, Section I, 7E
2021 commentThe function satisfies the differential equation
where and are complex analytic functions except, possibly, for isolated singularities in (the extended complex plane).
(a) Given equation , state the conditions for a point to be
(i) an ordinary point,
(ii) a regular singular point,
(iii) an irregular singular point.
(b) Now consider and use a suitable change of variables , with , to rewrite as a differential equation that is satisfied by . Hence, deduce the conditions for to be
(i) an ordinary point,
(ii) a regular singular point,
(iii) an irregular singular point.
[In each case, you should express your answer in terms of the functions and .]
(c) Use the results above to prove that any equation of the form ( ) must have at least one singular point in .
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Paper 3 , Section I, 7E
2021 commentThe Beta function is defined by
for and .
(a) Prove that and find .
(b) Show that .
(c) For each fixed with , use part (b) to obtain the analytic continuation of as an analytic function of , with the exception of the points
(d) Use part (c) to determine the type of singularity that the function has at , for fixed with .
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Paper 4 , Section I, 7E
2021 comment(a) Explain in general terms the meaning of the Papperitz symbol
State a condition satisfied by and . [You need not write down any differential equations explicitly, but should provide explicit explanation of the meaning of and
(b) The Papperitz symbol
where are constants, can be transformed into
(i) Provide an explicit description of the transformations required to obtain ( from .
(ii) One of the solutions to the -equation that corresponds to is a hypergeometric function . Express and in terms of and .
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Paper 1, Section II, E
2021 comment(a) Functions and are analytic in a connected open set with in a non-empty open subset . State the identity theorem.
(b) Let and be connected open sets with . Functions and are analytic on and respectively with on . Explain briefly what is meant by analytic continuation of and use part (a) to prove that analytic continuation to is unique.
(c) The function is defined by
where and is a positive integer. Use the method of contour deformation to construct the analytic continuation of into .
(d) The function is defined by
where and is a positive integer. Prove that experiences a discontinuity when crosses the real axis. Determine the value of this discontinuity. Hence, explain why cannot be used as an analytic continuation of .
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Paper 2, Section II, 13E
2021 commentThe temperature in a semi-infinite bar satisfies the heat equation
where is a positive constant.
For , the bar is at zero temperature. For , the temperature is subject to the boundary conditions
where and are positive constants, and as .
(a) Show that the Laplace transform of with respect to takes the form
and find . Hence write in terms of and .
(b) By performing the inverse Laplace transform using contour integration, show that for
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Paper 1, Section II, 18I
2021 comment(a) Let be fields, and a polynomial.
Define what it means for to be a splitting field for over .
Prove that splitting fields exist, and state precisely the theorem on uniqueness of splitting fields.
Let . Find a subfield of which is a splitting field for over Q. Is this subfield unique? Justify your answer.
(b) Let , where is a primitive 7 th root of unity.
Show that the extension is Galois. Determine all subfields .
For each subfield , find a primitive element for the extension explicitly in terms of , find its minimal polynomial, and write and .
Which of these subfields are Galois over ?
[You may assume the Galois correspondence, but should prove any results you need about cyclotomic extensions directly.]
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Paper 2, Section II, 18I
2021 comment(a) Let be a polynomial of degree , and let be its splitting field.
(i) Suppose that is irreducible. Compute , carefully stating any theorems you use.
(ii) Now suppose that factors as in , with each irreducible, and if . Compute , carefully stating any theorems you use.
(iii) Explain why is a cyclotomic extension. Define the corresponding homomorphism for this extension (for a suitable integer ), and compute its image.
(b) Compute for the polynomial . [You may assume that is irreducible and that its discriminant is .]
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Paper 3, Section II, 18I
2021 commentDefine the elementary symmetric functions in the variables . State the fundamental theorem of symmetric functions.
Let , where is a field. Define the discriminant of , and explain why it is a polynomial in .
Compute the discriminant of .
Let . When does the discriminant of equal zero? Compute the discriminant of .
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Paper 4 , Section II, 18I
2021 commentLet be a field, and a group which acts on by field automorphisms.
(a) Explain the meaning of the phrase in italics in the previous sentence.
Show that the set of fixed points is a subfield of .
(b) Suppose that is finite, and set . Let . Show that is algebraic and separable over , and that the degree of over divides the order of .
Assume that is a primitive element for the extension , and that is a subgroup of . What is the degree of over ? Justify your answer.
(c) Let , and let be a primitive th root of unity in for some integer . Show that the -automorphisms of defined by
generate a group isomorphic to the dihedral group of order .
Find an element for which .
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Paper 1, Section II, C
2021 commentThe Weyl tensor may be defined (in spacetime dimensions) as
where is the Riemann tensor, is the Ricci tensor and is the Ricci scalar.
(a) Show that and deduce that all other contractions vanish.
(b) A conformally flat metric takes the form
where is the Minkowski metric and is a scalar function. Calculate the Weyl tensor at a given point . [You may assume that at .]
(c) The Schwarzschild metric outside a spherically symmetric mass (such as the Sun, Earth or Moon) is
(i) Calculate the leading-order contribution to the Weyl component valid at large distances, , beyond the central spherical mass.
(ii) What physical phenomenon, known from ancient times, can be attributed to this component of the Weyl tensor at the location of the Earth? [This is after subtracting off the Earth's own gravitational field, and neglecting the Earth's motion within the solar system.] Briefly explain why your answer is consistent with the Einstein equivalence principle.
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Paper 2, Section II, 38C
2021 commentConsider the following metric for a 3-dimensional, static and rotationally symmetric Lorentzian manifold:
(a) Write down a Lagrangian for arbitrary geodesics in this metric, if the geodesic is affinely parameterized with respect to . What condition may be imposed to distinguish spacelike, timelike, and null geodesics?
(b) Find the three constants of motion for any geodesic.
(c) Two observation stations are sitting at radii and respectively, and at the same angular coordinate. Each is accelerating so as to remain stationary with respect to time translations. At a photon is emitted from the naked singularity at .
(i) At what time does the photon reach the inner station?
(ii) Express the frequency of the photon at the outer station in terms of the frequency at the inner station. Explain whether the photon is redshifted or blueshifted as it travels.
(d) Consider a complete (i.e. infinite in both directions) spacelike geodesic on a constant- slice with impact parameter . What is the angle between the two asymptotes of the geodesic at ? [You need not be concerned with the sign of or the periodicity of the coordinate.]
[Hint: You may find integration by substitution useful.]
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Paper 3, Section II, 37C
2021 comment(a) Determine the signature of the metric tensor given by
Is it Riemannian, Lorentzian, or neither?
(b) Consider a stationary black hole with the Schwarzschild metric:
These coordinates break down at the horizon . By making a change of coordinates, show that this metric can be converted to infalling Eddington-Finkelstein coordinates.
(c) A spherically symmetric, narrow pulse of radiation with total energy falls radially inwards at the speed of light from infinity, towards the origin of a spherically symmetric spacetime that is otherwise empty. Assume that the radial width of the pulse is very small compared to the energy , and the pulse can therefore be treated as instantaneous.
(i) Write down a metric for the region outside the pulse, which is free from coordinate singularities. Briefly justify your answer. For what range of coordinates is this metric valid?
(ii) Write down a metric for the region inside the pulse. Briefly justify your answer. For what range of coordinates is this metric valid?
(iii) What is the final state of the system?
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Paper 4 , Section II,
2021 comment(a) A flat , isotropic and homogeneous universe has metric given by
(i) Show that the non-vanishing Christoffel symbols and Ricci tensor components are
where dots are time derivatives and (no summation assumed).
(ii) Derive the first-order Friedmann equation from the Einstein equations,
(b) Consider a flat universe described by ( ) with in which late-time acceleration is driven by "phantom" dark energy obeying an equation of state with pressure , where and the energy density . The remaining matter is dust, so we have with each component separately obeying .
(i) Calculate an approximate solution for the scale factor that is valid at late times. Show that the asymptotic behaviour is given by a Big Rip, that is, a singularity in which at some finite time .
(ii) Sketch a diagram of the scale factor as a function of for a convenient choice of , ensuring that it includes (1) the Big Bang, (2) matter domination, (3) phantom-energy domination, and (4) the Big Rip. Label these epochs and mark them on the axes.
(iii) Most reasonable classical matter fields obey the null energy condition, which states that the energy-momentum tensor everywhere satisfies for any null vector . Determine if this applies to phantom energy.
[The energy-momentum tensor for a perfect fluid is
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Paper 1, Section II, 17G
2021 commentDefine the binomial random graph , where and .
(a) Let and let be the event that contains a copy of the complete graph . Show that if is such that then as .
(b) State Chebyshev's inequality. Show that if then .
(c) Let be a triangle with an added leaf vertex, that is
where are distinct. Let be the event that contains a copy of . Show that if then .
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Paper 2, Section II, 17 G
2021 comment(a) Define a tree and what it means for a graph to be acyclic. Show that if is an acyclic graph on vertices then . [You may use the fact that a spanning tree on vertices has edges.]
(b) Show that any 3-regular graph on vertices contains a cycle of length . Hence show that there exists such that every 3-regular graph on more than vertices must contain two cycles with disjoint vertex sets.
(c) An unfriendly partition of a graph is a partition , where , such that every vertex has and every has . Show that every graph with has an unfriendly partition.
(d) A friendly partition of a graph is a partition , where , such that every vertex has and every has . Give an example of a 3-regular graph (on at least 1 vertex) that does not have a friendly partition. Using part (b), show that for large enough every 3-regular graph with has a friendly partition.
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Paper 3, Section II, 17G
2021 comment(a) Define the Ramsey number and show that .
Show that every 2-coloured complete graph with contains a monochromatic spanning tree. Is the same true if is coloured with 3 colours? Give a proof or counterexample.
(b) Let be a graph. Show that the number of paths of length 2 in is
Now consider a 2-coloured complete graph with . Show that the number of monochromatic triangles in is
where denotes the number of red edges incident with a vertex and denotes the number of blue edges incident with . [Hint: Count paths of length 2 in two different ways.]
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Paper 4, Section II, 17G
2021 commentState and prove Hall's theorem, giving any definitions required by the proof (e.g. of an -alternating path).
Let be a (not necessarily bipartite) graph, and let be the size of the largest matching in . Let be the smallest for which there exist vertices such that every edge in is incident with at least one of . Show that and that . For each positive integer , find a graph with and . Determine and when is the Turan graph on 30 vertices.
By using Hall's theorem, or otherwise, show that if is a bipartite graph then
Define the chromatic index of a graph . Prove that if with then .
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Paper 1 , Section II, 33D
2021 comment(a) Let and be matrix-valued functions, whilst is a vector-valued function. Show that the linear system
is over-determined and derive a consistency condition on that is necessary for there to be non-trivial solutions.
(b) Suppose that
where is a scalar function. Obtain a partial differential equation for that is equivalent to your consistency condition from part (a).
(c) Now let and suppose is independent of . Show that the trace of is constant for all positive integers . Hence, or otherwise, construct a non-trivial first integral of the equation
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Paper 2, Section II, 34D
2021 comment(a) Explain briefly how the linear operators and can be used to give a Lax-pair formulation of the equation .
(b) Give a brief definition of the scattering data
attached to a smooth solution of the KdV equation at time . [You may assume to be rapidly decreasing in .] State the time dependence of and , and derive the time dependence of from the Lax-pair formulation.
(c) Show that
satisfies . Now let be the solution of the equation
and let , where . Defining by , show that
(d) Given that obeys the equations
where , deduce that
and hence that solves the equation.
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Paper 3, Section II, 32D
2021 comment(a) Consider the group of transformations of given by , where . Show that this acts as a group of Lie symmetries for the equation .
(b) Let and define . Show that the vector field generates the group of phase rotations .
(c) Show that the transformations of defined by
form a one-parameter group generated by the vector field
and find the second prolongation of the action of . Hence find the coefficients and in the second prolongation of ,
complex conjugate .
(d) Show that the group of transformations in part (c) acts as a group of Lie symmetries for the nonlinear Schrödinger equation . Given that solves the nonlinear Schrödinger equation for any , find a solution which describes a solitary wave travelling at arbitrary speed .
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Paper 1, Section II, 22H
2021 commentLet be a separable Hilbert space and be a Hilbertian (orthonormal) basis of . Given a sequence of elements of and , we say that weakly converges to , denoted , if .
(a) Given a sequence of elements of , prove that the following two statements are equivalent:
(i) such that ;
(ii) the sequence is bounded in and , the sequence is convergent.
(b) Let be a bounded sequence of elements of . Show that there exists and a subsequence such that in .
(c) Let be a sequence of elements of and be such that . Show that the following three statements are equivalent:
(i) ;
(ii) ;
(iii) such that .
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Paper 2, Section II, 22H
2021 comment(a) Let be a real normed vector space. Show that any proper subspace of has empty interior.
Assuming to be infinite-dimensional and complete, prove that any algebraic basis of is uncountable. [The Baire category theorem can be used if stated properly.] Deduce that the vector space of polynomials with real coefficients cannot be equipped with a complete norm, i.e. a norm that makes it complete.
(b) Suppose that and are norms on a vector space such that and are both complete. Prove that if there exists such that for all , then there exists such that for all . Is this still true without the assumption that and are both complete? Justify your answer.
(c) Let be a real normed vector space (not necessarily complete) and be the set of linear continuous forms . Let be a sequence in such that for all . Prove that
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Paper 3, Section II, H
2021 comment(a) State the Arzela-Ascoli theorem, including the definition of equicontinuity.
(b) Consider a sequence of continuous real-valued functions on such that for all is bounded and the sequence is equicontinuous at . Prove that there exists and a subsequence such that uniformly on any closed bounded interval.
(c) Let be a Hausdorff compact topological space, and the real-valued continuous functions on . Let be a compact subset of . Prove that the collection of functions is equicontinuous.
(d) We say that a Hausdorff topological space is locally compact if every point has a compact neighbourhood. Let be such a space, compact and open such that . Prove that there exists continuous with compact support contained in and equal to 1 on . [Hint: Construct an open set such that and is compact, and use Urysohn's lemma to construct a function in and then extend it by zero.]
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Paper 4, Section II, H
2021 comment(a) Let be two Hilbert spaces, and be a bounded linear operator. Show that there exists a unique bounded linear operator such that
(b) Let be a separable Hilbert space. We say that a sequence is a frame of if there exists such that
State briefly why such a frame exists. From now on, let be a frame of . Show that is dense in .
(c) Show that the linear map given by is bounded and compute its adjoint .
(d) Assume now that is a Hilbertian (orthonormal) basis of and let . Show that the Hilbert cube such that is a compact subset of .
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Paper 1, Section II, 16G
2021 commentLet and be sets of propositional formulae.
(a) What does it mean to say that is deductively closed? What does it mean to say that is consistent? Explain briefly why if is inconsistent then some finite subset of is inconsistent.
(b) We write to mean for all . If and we say and are equivalent. If is equivalent to a finite set of formulae we say that is finitary. Show that if is finitary then there is a finite set with .
(c) Now let be deductively closed sets of formulae with
Show that each is consistent.
Let . Show that is consistent and deductively closed, but that it is not finitary.
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Paper 2, Section II, G
2021 commentWrite down the inductive definition of ordinal exponentiation. Show that for every ordinal . Deduce that, for every ordinal , there is a least ordinal with . Show that, if , then must be a successor ordinal.
Now let be a non-zero ordinal. Show that there exist ordinals and , where , and a positive integer such that . Hence, or otherwise, show that can be written in the form
where are positive integers and are ordinals. [We call this the Cantor normal form of , and you may henceforth assume that it is unique.]
Given ordinals and positive integers find the Cantor normal form of . Hence, or otherwise, given non-zero ordinals and , find the Cantor normal form of in terms of the Cantor normal forms
and
of and .
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Paper 3, Section II, 16G
2021 comment(a) Let and be cardinals. What does it mean to say that ? Explain briefly why, assuming the Axiom of Choice, every infinite cardinal is of the form for some ordinal , and that for every ordinal we have .
(b) Henceforth, you should not assume the Axiom of Choice.
Show that, for any set , there is an injection from to its power set , but there is no bijection from to . Deduce that if is a cardinal then .
Let and be sets, and suppose that there exists a surjection . Show that there exists an injection .
Let be an ordinal. Prove that .
By considering as the set of relations on , or otherwise, show that there exists a surjection . Deduce that .
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Paper 4, Section II, 16G
2021 commentWrite down the Axiom of Foundation.
What is the transitive closure of a set ? Prove carefully that every set has a transitive closure. State and prove the principle of -induction.
Let be a model of . Let be a surjective function class such that for all we have if and only if . Show, by -induction or otherwise, that is the identity.
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Paper 1, Section I, 6E
2021 comment(a) Consider a population of size whose per capita rates of birth and death are and , respectively, where and all parameters are positive constants.
(i) Write down the equation for the rate of change of the population.
(ii) Show that a population of size is stationary and that it is asymptotically stable.
(b) Consider now a disease introduced into this population, where the number of susceptibles and infectives, and , respectively, satisfy the equations
(i) Interpret the biological meaning of each term in the above equations and comment on the reproductive capacity of the susceptible and infected individuals.
(ii) Show that the disease-free equilibrium, and , is linearly unstable if
(iii) Show that when the disease-free equilibrium is unstable there exists an endemic equilibrium satisfying
and that this equilibrium is linearly stable.
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Paper 2, Section I, E
2021 commentConsider a stochastic birth-death process in a population of size , where deaths occur in pairs for . The probability per unit time of a birth, for , is , that of a pair of deaths, for , is , and that of the death of a lonely singleton, , is .
(a) Write down the master equation for , the probability of a population of size at time , distinguishing between the cases and .
(b) For a function , show carefully that
where .
(c) Deduce the evolution equation for the mean , and simplify it for the case .
(d) For the same value of , show that
Deduce that the variance in the stationary state for satisfies
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Paper 3, Section I, E
2021 commentThe population density of individuals of age at time satisfies the partial differential equation
with the boundary condition
where and are, respectively, the per capita age-dependent birth and death rates.
(a) What is the biological interpretation of the boundary condition?
(b) Solve equation (1) assuming a separable form of solution, .
(c) Use equation (2) to obtain a necessary condition for the existence of a separable solution to the full problem.
(d) For a birth rate with and an age-independent death rate , show that a separable solution to the full problem exists and find the critical value of above which the population density grows with time.
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Paper 4, Section I, E
2021 commentA marine population grows logistically and disperses by diffusion. It is moderately predated on up to a distance from a straight coast. Beyond that distance, predation is sufficiently excessive to eliminate the population. The density of the population at a distance from the coast satisfies
subject to the boundary conditions
(a) Interpret the terms on the right-hand side of , commenting on their dependence on . Interpret the boundary conditions.
(b) Show that a non-zero population is viable if and
Interpret these conditions.
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Paper 3, Section II, 13E
2021 commentConsider an epidemic spreading in a population that has been aggregated by age into groups numbered . The th age group has size and the numbers of susceptible, infective and recovered individuals in this group are, respectively, and . The spread of the infection is governed by the equations
where
and is a matrix satisfying , for .
(a) Describe the biological meaning of the terms in equations (1) and (2), of the matrix and the condition it satisfies, and of the lack of dependence of and on .
State the condition on the matrix that would ensure the absence of any transmission of infection between age groups.
(b) In the early stages of an epidemic, and . Use this information to linearise the dynamics appropriately, and show that the linearised system predicts
where is the vector of infectives at time is the identity matrix and is a matrix that should be determined.
(c) Deduce a condition on the eigenvalues of the matrix that allows the epidemic to grow.
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Paper 4, Section II, E
2021 commentThe spatial density of a population at location and time satisfies
where and .
(a) Give a biological example of the sort of phenomenon that this equation describes.
(b) Show that there are three spatially homogeneous and stationary solutions to , of which two are linearly stable to homogeneous perturbations and one is linearly unstable.
(c) For , find the stationary solution to subject to the conditions
(d) Write down the differential equation that is satisfied by a travelling-wave solution to of the form . Let be the solution from part (c). Verify that satisfies this differential equation for , provided the speed is chosen appropriately. [Hint: Consider the change to the equation from part (c).]
(e) State how the sign of depends on , and give a brief qualitative explanation for why this should be the case.
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Paper 1, Section II, J
2021 commentLet be a family of functions with . Define the shattering coefficient and the dimension of .
Briefly explain why if and , then .
Prove that if is a vector space of functions with and we define
then .
Let be the set of all spheres in . Suppose . Show that
Hint: Consider the class of functions , where
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Paper 1, Section II, 20G
2021 commentLet , where
(a) Show that .
(b) Let . By considering the matrix of acting on by multiplication, or otherwise, show that is an algebraic integer, and that is a -basis for [The discriminant of is , and 307 is prime.]
(c) Compute the prime factorisation of the ideal (3) in . Is (2) a prime ideal of Justify your answer.
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Paper 2, Section II, J
2021 comment(a) What is meant by the subdifferential of a convex function at ? Write down the subdifferential of the function given by , where .
Show that minimises if and only if .
What does it mean for a function to be strictly convex? Show that any minimiser of a strictly convex function must be unique.
(b) Suppose we have input-output pairs with . Consider the objective function
where and . Assume that . Fix and define
where for . Show that if , then
[You may use any results from the course without proof, other than those whose proof is asked for directly.]
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Paper 4, Section II, J
2021 commentLet be a dataset of input-output pairs lying in for . Describe the random-forest algorithm as applied to using decision trees to produce a fitted regression function . [You need not explain in detail the construction of decision trees, but should describe any modifications specific to the random-forest algorithm.]
Briefly explain why for each and , we have .
State the bounded-differences inequality.
Treating as deterministic, show that with probability at least ,
where .
Hint: Treat each as a random variable taking values in an appropriate space (of functions), and consider a function satisfying
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Paper 2, Section II, 20G
2021 commentLet be a field containing . What does it mean to say that an element of is algebraic? Show that if is algebraic and non-zero, then there exists such that is a non-zero (rational) integer.
Now let be a number field, with ring of integers . Let be a subring of whose field of fractions equals . Show that every element of can be written as , where and is a positive integer.
Prove that is a free abelian group of , and that has finite index in . Show also that for every nonzero ideal of , the index of in is finite, and that for some positive integer is an ideal of .
Suppose that for every pair of non-zero ideals , we have
Show that .
[You may assume without proof that is a free abelian group of rank ] ]
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Paper 4 , Section II, 20G
2021 comment(a) Compute the class group of . Find also the fundamental unit of , stating clearly any general results you use.
[The Minkowski bound for a real quadratic field is ]
(b) Let be real quadratic, with embeddings . An element is totally positive if and . Show that the totally positive elements of form a subgroup of the multiplicative group of index 4 .
Let be non-zero ideals. We say that is narrowly equivalent to if there exists a totally positive element of such that . Show that this is an equivalence relation, and that the equivalence classes form a group under multiplication. Show also that the order of this group equals
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Paper 1, Section I, 1I
2021 commentState Euler's criterion.
Let be an odd prime. Show that every primitive root modulo is a quadratic non-residue modulo .
Let be a Fermat prime, that is, a prime of the form for some . By evaluating , or otherwise, show that every quadratic non-residue modulo is a primitive root modulo . Deduce that 3 is a primitive root modulo for every Fermat prime .
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Paper 2, Section , I
2021 commentDefine the Möbius function , and explain what it means for it to be multiplicative.
Show that for every positive integer
where is the Euler totient function.
Fix an integer . Use the Chinese remainder theorem to show that there are infinitely many positive integers for which
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Paper 3, Section I, I
2021 commentDefine the continued fraction expansion of , and show that this expansion terminates if and only if .
Define the convergents of the continued fraction expansion of , and show that for all ,
Deduce that if , then for all , at least one of
must hold.
[You may assume that lies strictly between and for all ]
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Paper 4, Section I, I
2021 commentLet be a prime, and let for some positive integer .
Show that if a prime power divides for some , then .
Given a positive real , define , where is the von Mangoldt function, taking the value if for some prime and integer , and 0 otherwise. Show that
Deduce that for all integers .
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Paper 3, Section II, I
2021 commentState what it means for two binary quadratic forms to be equivalent, and define the class number .
Let be a positive integer, and let be a binary quadratic form. Show that properly represents if and only if is equivalent to a binary quadratic form
for some integers and .
Let be an integer such that or . Show that is properly represented by some binary quadratic form of discriminant if and only if is a square modulo .
Fix a positive integer . Show that is composite for some integer such that if and only if is a square modulo for some prime .
Deduce that if and only if is prime for all .
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Paper 4, Section II, I
2021 comment(a) Let be an odd integer and an integer with . What does it mean to say that is a (Fermat) pseudoprime to base b?
Let be integers. Show that if is an odd composite integer dividing and satisfying , then is a pseudoprime to base .
(b) Fix . Let be an odd prime not dividing , and let
Use the conclusion of part (a) to show that is a pseudoprime to base . Deduce that there are infinitely many pseudoprimes to base .
(c) Let be integers, and let , where are distinct primes not dividing . For each , let . Show that is a pseudoprime to base if and only if for all , the order of modulo divides .
(d) By considering products of prime factors of and for primes , deduce that there are infinitely many pseudoprimes to base 2 with two prime factors.
[Hint: You may assume that for implies , and that for is not a power of 3.]
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Paper 1, Section II, E
2021 commentLet with and define is not invertible .
The QR algorithm for computing is defined as follows. Set . For compute the factorization and set . (Here is an orthogonal matrix and is an upper triangular matrix.)
(a) Show that is related to the original matrix by the similarity transformation , where is orthogonal and is the QR factorization of with .
(b) Suppose that is symmetric and that its eigenvalues satisfy
Suppose, in addition, that the first two canonical basis vectors are given by , , where for and are the normalised eigenvectors of .
Let be the upper left corner of . Show that as , where and denotes the Hausdorff metric
[Hint: You may use the fact that for real symmetric matrices we have
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Paper 2, Section II, 41E
2021 comment(a) Let and define by
Let be defined as the discrete Fourier transform (DFT) of , i.e.
Show that
(b) Define the discrete cosine transform by
For with , show that, similar to the Fast Fourier Transform (FFT), there exists an algorithm that computes the DCT of a vector of length , where the number of multiplications required is bounded by , where is some constant independent of .
[You may not assume that the FFT algorithm requires multiplications to compute the DFT of a vector of length . If you use this, you must prove it.]
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Paper 3, Section II, 40E
2021 commentConsider discretisation of the diffusion equation
by the Crank-Nicholson method:
where is the Courant number, is the step size in the space discretisation, is the step size in the time discretisation, and , where is the solution of . The initial condition is given.
(a) Consider the Cauchy problem for on the whole line, (thus ), and derive the formula for the amplification factor of the Crank-Nicholson method ( ). Use the amplification factor to show that the Crank-Nicholson method is stable for the Cauchy problem for all .
[You may quote basic properties of the Fourier transform mentioned in lectures, but not the theorem on sufficient and necessary conditions on the amplification factor to have stability.]
(b) Consider on the interval (thus and ) with Dirichlet boundary conditions and , for some sufficiently smooth functions and . Show directly (without using the Lax equivalence theorem) that, given sufficient smoothness of , the Crank-Nicholson method is convergent, for any , in the norm defined by for .
[You may assume that the Trapezoidal method has local order 3 , and that the standard three-point centred discretisation of the second derivative (as used in the CrankNicholson method) has local order 2.]
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Paper 4 , Section II, 40E
2021 comment(a) Show that if and are real matrices such that both and are symmetric positive definite, then the spectral radius of is strictly less than
(b) Consider the Poisson equation (with zero Dirichlet boundary condition) on the unit square, where is some smooth function. Given and an equidistant grid on the unit square with stepsize , the standard five-point method is given by
where and . Equation can be written as a linear system , where and both depend on the chosen ordering of the grid points.
Use the result in part (a) to show that the Gauss-Seidel method converges for the linear system described above, regardless of the choice of ordering of the grid points.
[You may quote convergence results - based on the spectral radius of the iteration matrix - mentioned in the lecture notes.]
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Paper 1, Section II, B
2021 comment(a) A group of transformations acts on a quantum system. Briefly explain why the Born rule implies that these transformations may be represented by operators obeying
for all , where .
What additional property does have when is a group of symmetries of the Hamiltonian? Show that symmetries correspond to conserved quantities.
(b) The Coulomb Hamiltonian describing the gross structure of the hydrogen atom is invariant under time reversal, . Suppose we try to represent time reversal by a unitary operator obeying , where is the time-evolution operator. Show that this would imply that hydrogen has no stable ground state.
An operator is antilinear if
for all and all , and antiunitary if, in addition,
where and . Show that if time reversal is instead represented by an antiunitary operator then the above instability of hydrogen is avoided.
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Paper 2, Section II, B
2021 comment(a) Let be a basis of eigenstates of a non-degenerate Hamiltonian , with corresponding eigenvalues . Write down an expression for the energy levels of the perturbed Hamiltonian , correct to second order in the dimensionless constant .
(b) A particle travels in one dimension under the influence of the potential
where is the mass, a frequency and a length scale. Show that, to first order in , all energy levels coincide with those of the harmonic oscillator. Calculate the energy of the ground state to second order in .
Does perturbation theory in converge for this potential? Briefly explain your answer.
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Paper 3, Section II, B
2021 comment(a) A quantum system with total angular momentum is combined with another of total angular momentum . What are the possible values of the total angular momentum of the combined system? For given , what are the possible values of the angular momentum along any axis?
(b) Consider the case . Explain why all the states with are antisymmetric under exchange of the angular momenta of the two subsystems, while all the states with are symmetric.
(c) An exotic particle of spin 0 and negative intrinsic parity decays into a pair of indistinguishable particles . Assume each particle has spin 1 and that the decay process conserves parity. Find the probability that the direction of travel of the particles is observed to lie at an angle from some axis along which their total spin is observed to be ?
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Paper 4, Section II, 33B
2021 comment(a) A quantum system has Hamiltonian . Let be an orthonormal basis of eigenstates, with corresponding energies . For , and the system is in state . Calculate the probability that it is found to be in state at time , correct to lowest non-trivial order in .
(b) Now suppose form a basis of the Hilbert space, with respect to which
where is the Heaviside step function and is a real constant. Calculate the exact probability that the system is in state at time . For which frequency is this probability maximized?
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Paper 1, Section II, J
2021 commentLet be random variables with joint probability density function in a statistical model .
(a) Define the Fisher information . What do we mean when we say that the Fisher information tensorises?
(b) Derive the relationship between the Fisher information and the derivative of the score function in a regular model.
(c) Consider the model defined by and
where are i.i.d. random variables, and is a known constant. Compute the Fisher information . For which values of does the Fisher information tensorise? State a lower bound on the variance of an unbiased estimator in this model.
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Paper 2, Section II, J
2021 commentLet be i.i.d. random observations taking values in with a continuous distribution function . Let for each .
(a) State the Kolmogorov-Smirnov theorem. Explain how this theorem may be used in a goodness-of-fit test for the null hypothesis , with continuous.
(b) Suppose you do not have access to the quantiles of the sampling distribution of the Kolmogorov-Smirnov test statistic. However, you are given i.i.d. samples with distribution function . Describe a test of with size exactly .
(c) Now suppose that are i.i.d. taking values in with probability density function , with . Define the density estimator
Show that for all and all ,
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Paper 3, Section II, J
2021 commentLet iid for some known and some unknown . [The gamma distribution has probability density function
and its mean and variance are and , respectively.]
(a) Find the maximum likelihood estimator for and derive the distributional limit of . [You may not use the asymptotic normality of the maximum likelihood estimator proved in the course.]
(b) Construct an asymptotic -level confidence interval for and show that it has the correct (asymptotic) coverage.
(c) Write down all the steps needed to construct a candidate to an asymptotic -level confidence interval for using the nonparametric bootstrap.
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Paper 4, Section II, J
2021 commentSuppose that , and suppose the prior on is a gamma distribution with parameters and . [Recall that has probability density function
and that its mean and variance are and , respectively. ]
(a) Find the -Bayes estimator for for the quadratic loss, and derive its quadratic risk function.
(b) Suppose we wish to estimate . Find the -Bayes estimator for for the quadratic loss, and derive its quadratic risk function. [Hint: The moment generating function of a Poisson distribution is for , and that of a Gamma distribution is for .]
(c) State a sufficient condition for an admissible estimator to be minimax, and give a proof of this fact.
(d) For each of the estimators in parts (a) and (b), is it possible to deduce using the condition in (c) that the estimator is minimax for some value of and ? Justify your answer.
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Paper 1, Section II, H
2021 comment(a) State and prove Fatou's lemma. [You may use the monotone convergence theorem without proof, provided it is clearly stated.]
(b) Show that the inequality in Fatou's lemma can be strict.
(c) Let and be non-negative random variables such that almost surely as . Must we have ?
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Paper 2, Section II, H
2021 commentLet be a measure space. A function is simple if it is of the form , where and .
Now let be a Borel-measurable map. Show that there exists a sequence of simple functions such that for all as .
Next suppose is also -integrable. Construct a sequence of simple -integrable functions such that as .
Finally, suppose is also bounded. Show that there exists a sequence of simple functions such that uniformly on as .
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Paper 3, Section II,
2021 commentShow that random variables defined on some probability space are independent if and only if
for all bounded measurable functions .
Now let be an infinite sequence of independent Gaussian random variables with zero means, , and finite variances, . Show that the series converges in if and only if .
[You may use without proof that for .]
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Paper 4, Section II, 26H
2021 commentLet be a probability space. Show that for any sequence satisfying one necessarily has
Let and be random variables defined on . Show that almost surely as implies that in probability as .
Show that in probability as if and only if for every subsequence there exists a further subsequence such that almost surely as .
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Paper 1, Section I,
2021 commentAlice wishes to communicate to Bob a 1-bit message or chosen by her with equal prior probabilities . For (respectively ) she sends Bob the quantum state (respectively ). On receiving the state, Bob applies quantum operations to it, to try to determine Alice's message. The Helstrom-Holevo theorem asserts that the probability for Bob to correctly determine Alice's message is bounded by , where , and that this bound is achievable.
(a) Suppose that and , and that Bob measures the received state in the basis , where and , to produce his output 0 or 1 , respectively. Calculate the probability that Bob correctly determines Alice's message, and show that the maximum value of over choices of achieves the Helstrom-Holevo bound.
(b) State the no-cloning theorem as it applies to unitary processes and a set of two non-orthogonal states . Show that the Helstrom-Holevo theorem implies the validity of the no-cloning theorem in this situation.
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Paper 2, Section I, 10D
2021 commentLet denote the set of all -bit strings and let be a Boolean function which obeys either
(I) for all , or
(II) for exactly half of all .
Suppose we are given the -qubit state
Show how we may determine with certainty whether is of case (I) or case (II).
Suppose now that Alice and Bob are separated in space. Alice possesses a quantum oracle for a Boolean function and Bob similarly possess a quantum oracle for a Boolean function . These functions are arbitrary, except that either
(1) for all , or
(2) for exactly half of all .
Alice and Bob each have available a supply of qubits in state and each can apply local quantum operations (including their own function oracle) to any qubits in their possession. Additionally, they can send qubits to each other.
Show how Bob may decide with certainty which case applies, after he has received qubits from Alice. [Hint: You may find it helpful to consider the function , where denotes addition mod 2.]
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Paper 3 , Section I, D
2021 commentLet be the joint state of a bipartite system with subsystems and separated in space. Suppose that Alice and Bob have access only to subsystems and respectively, on which they can perform local quantum operations.
Alice performs a unitary operation on and then a (generally incomplete) measurement on , with projectors labelled by her possible measurement outcomes . Then Bob performs a complete measurement on relative to the orthonormal basis labelled by his possible outcomes .
Show that the probability distribution of Bob's measurement outcomes is unaffected by whether or not Alice actually performs the local operations on described above.
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Paper 4, Section I,
2021 commentLet be a state space of dimension with standard orthonormal basis labelled by . Let QFT denote the quantum Fourier transform and let denote the operation defined by .
(a) Introduce the basis defined by . Show that each is an eigenstate of and determine the corresponding eigenvalue.
(b) By expressing a generic state in the basis, show that QFT and QFT have the same output distribution if measured in the standard basis.
(c) Let be positive integers with , and let be an integer with . Suppose that we are given the state
where and are unknown to us. Using part (b) or otherwise, show that a standard basis measurement on QFT has an output distribution that is independent of .
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Paper 2, Section II, D
2021 commentAlice and Bob are separated in space and can communicate only over a noiseless public classical channel, i.e. they can exchange bit string messages perfectly, but the messages can be read by anyone. An eavesdropper Eve constantly monitors the channel, but cannot alter any passing messages. Alice wishes to communicate an -bit string message to Bob whilst keeping it secret from Eve.
(a) Explain how Alice can do this by the one-time pad method, specifying clearly any additional resource that Alice and Bob need. Explain why in this method, Alice's message does, in fact, remain secure against eavesdropping.
(b) Suppose now that Alice and Bob do not possess the additional resource needed in part (a) for the one-time pad, but that they instead possess pairs of qubits, where , with each pair being in the state
where the real parameters are known to Alice and Bob and obey and . For each qubit pair in state , Alice possesses qubit and Bob possesses qubit . They each also have available a supply of ancilla qubits, each in state , and they can each perform local quantum operations on qubits in their possession.
Show how Alice, using only local quantum operations, can convert each state into by a process that succeeds with non-zero probability. [Hint: It may be useful for Alice to start by adjoining an ancilla qubit and work locally on her two qubits in
Hence, or otherwise, show how Alice can communicate a bit string of expected length to Bob in a way that keeps it secure against eavesdropping by Eve.
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Paper 3, Section II, D
2021 commentLet denote the set of all -bit strings and let denote the space of qubits.
(a) Suppose has the property that for a unique and suppose we have a quantum oracle .
(i) Let and introduce the operators
on , where is the identity operator. Give a geometrical description of the actions of and on the 2-dimensional subspace of given by the real span of and . [You may assume without proof that the product of two reflections in is a rotation through twice the angle between the mirror lines.]
(ii) Using the results of part (i), or otherwise, show how we may determine with certainty, starting with a supply of qubits each in state and using only once, together with other quantum operations that are independent of .
(b) Suppose , where is a fixed linear subspace with orthogonal complement . Let denote the projection operator onto and let , where is the identity operator on .
(i) Show that any can be written as , where , and and are normalised.
(ii) Let and . Show that .
(iii) Now assume, in addition, that and that for some unitary operation . Suppose we can implement the operators as well as the operation . In the case , show how the -qubit state may be made exactly from by a process that succeeds with certainty.
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Paper 1, Section II, I
2021 comment(a) What does it mean to say that a representation of a group is completely reducible? State Maschke's theorem for representations of finite groups over fields of characteristic 0 . State and prove Schur's lemma. Deduce that if there exists a faithful irreducible complex representation of , then is cyclic.
(b) If is any finite group, show that the regular representation is faithful. Show further that for every finite simple group , there exists a faithful irreducible complex representation of .
(c) Which of the following groups have a faithful irreducible representation? Give brief justification of your answers.
(i) the cyclic groups a positive integer ;
(ii) the dihedral group ;
(iii) the direct product .
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Paper 2, Section II, I
2021 commentLet be a finite group and work over .
(a) Let be a faithful character of , and suppose that takes precisely different values as varies over all the elements of . Show that every irreducible character of is a constituent of one of the powers . [Standard properties of the Vandermonde matrix may be assumed if stated correctly.]
(b) Assuming that the number of irreducible characters of is equal to the number of conjugacy classes of , show that the irreducible characters of form a basis of the complex vector space of all class functions on . Deduce that are conjugate if and only if for all characters of .
(c) Let be a character of which is not faithful. Show that there is some irreducible character of such that for all integers .
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Paper 3, Section II, I
2021 commentIn this question we work over .
(a) (i) Let be a subgroup of a finite group . Given an -space , define the complex vector space . Define, with justification, the -action on .
(ii) Write for the conjugacy class of . Suppose that breaks up into conjugacy classes of with representatives . If is a character of , write down, without proof, a formula for the induced character as a certain sum of character values .
(b) Define permutations by and let be the subgroup of . It is given that the elements of are all of the form for and that has order 21 .
(i) Find the orders of the centralisers and . Hence show that there are five conjugacy classes of .
(ii) Find all characters of degree 1 of by lifting from a suitable quotient group.
(iii) Let . By first inducing linear characters of using the formula stated in part (a)(ii), find the remaining irreducible characters of .
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Paper 4, Section II, I
2021 comment(a) Define the group . Sketch a proof of the classification of the irreducible continuous representations of . Show directly that the characters obey an orthogonality relation.
(b) Define the group .
(i) Show that there is a bijection between the conjugacy classes in and the subset of the real line. [If you use facts about a maximal torus , you should prove them.]
(ii) Write for the conjugacy class indexed by an element , where . Show that is homeomorphic to . [Hint: First show that is in bijection with .
(iii) Let be the parametrisation of conjugacy classes from part (i). Determine the representation of whose character is the function .
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Paper 1, Section II, F
2021 comment(a) Consider an open . Prove that a real-valued function is harmonic if and only if
for some analytic function .
(b) Give an example of a domain and a harmonic function that is not equal to the real part of an analytic function on . Justify your answer carefully.
(c) Let be a harmonic function on such that for every . Prove that is constant, justifying your answer carefully. Exhibit a countable subset and a non-constant harmonic function on such that for all we have and .
(d) Prove that every non-constant harmonic function is surjective.
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Paper 2, Section II, F
2021 commentLet be a domain, let be a function element in , and let be a path with . Define what it means for a function element to be an analytic continuation of along .
Suppose that is a path homotopic to and that is an analytic continuation of along . Suppose, furthermore, that can be analytically continued along any path in . Stating carefully any theorems that you use, prove that .
Give an example of a function element that can be analytically continued to every point of and a pair of homotopic paths in starting in such that the analytic continuations of along and take different values at .
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Paper 3, Section II, F
2021 comment(a) Let be a polynomial of degree , and let be the multiplicities of the ramification points of . Prove that
Show that, for any list of integers satisfying , there is a polynomial of degree such that the are the multiplicities of the ramification points of .
(b) Let be an analytic map, and let be the set of branch points. Prove that the restriction is a regular covering map. Given , explain how a closed loop in gives rise to a permutation of . Show that the group of all such permutations is transitive, and that the permutation only depends on up to homotopy.
(c) Prove that there is no meromorphic function of degree 4 with branch points such that every preimage of 0 and 1 has ramification index 2 , while some preimage of has ramification index equal to 3. [Hint: You may use the fact that every non-trivial product of -cycles in the symmetric group is a -cycle.]
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Paper 1, Section I, J
2021 commentLet . The probability density function of the inverse Gaussian distribution (with the shape parameter equal to 1 ) is given by
Show that this is a one-parameter exponential family. What is its natural parameter? Show that this distribution has mean and variance .
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Paper 2, Section I, J
2021 commentDefine a generalised linear model for a sample of independent random variables. Define further the concept of the link function. Define the binomial regression model (without the dispersion parameter) with logistic and probit link functions. Which of these is the canonical link function?
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Paper 3, Section I, J
2021 commentConsider the normal linear model , where is a design matrix, is a vector of responses, is the identity matrix, and are unknown parameters.
Derive the maximum likelihood estimator of the pair and . What is the distribution of the estimator of ? Use it to construct a -level confidence interval of . [You may use without proof the fact that the "hat matrix" is a projection matrix.]
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Paper 4 , Section I, J
2021 commentThe data frame data contains the daily number of new avian influenza cases in a large poultry farm.

Write down the model being fitted by the code below. Does the model seem to provide a satisfactory fit to the data? Justify your answer.
The owner of the farm estimated that the size of the epidemic was initially doubling every 7 days. Is that estimate supported by the analysis below? [You may need .]


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Paper 1, Section II, J
2021 commentThe following data were obtained in a randomised controlled trial for a drug. Due to a manufacturing error, a subset of trial participants received a low dose (LD) instead of a standard dose (SD) of the drug.

(a) Below we analyse the data using Poisson regression:

(i) After introducing necessary notation, write down the Poisson models being fitted above.
(ii) Write down the corresponding multinomial models, then state the key theoretical result (the "Poisson trick") that allows you to fit the multinomial models using Poisson regression. [You do not need to prove this theoretical result.]
(iii) Explain why the number of degrees of freedom in the likelihood ratio test is 2 in the analysis of deviance table. What can you conclude about the drug?
(b) Below is the summary table of the second model:

(i) Drug efficacy is defined as one minus the ratio of the probability of worsening in the treated group to the probability of worsening in the control group. By using a more sophisticated method, a published analysis estimated that the drug efficacy is for the LD treatment and for the treatment. Are these numbers similar to what is obtained by Poisson regression? [Hint: , and , where is the base of the natural logarithm.]
(ii) Explain why the information in the summary table is not enough to test the hypothesis that the LD drug and the SD drug have the same efficacy. Then describe how you can test this hypothesis using analysis of deviance in .
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Paper 4, Section II, J
2021 commentLet be an non-random design matrix and be a -vector of random responses. Suppose , where is an unknown vector and is known.
(a) Let be a constant. Consider the ridge regression problem
Let be the fitted values. Show that , where
(b) Show that
(c) Let , where is independent of . Show that is an unbiased estimator of .
(d) Describe the behaviour (monotonicity and limits) of as a function of when and . What is the minimum value of ?
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Paper 1, Section II, 36C
2021 commentThroughout this question you should consider a classical gas and assume that the number of particles is fixed.
(a) Write down the equation of state for an ideal gas. Write down an expression for the internal energy of an ideal gas in terms of the heat capacity at constant volume, .
(b) Starting from the first law of thermodynamics, find a relation between and the heat capacity at constant pressure, , for an ideal gas. Hence give an expression for .
(c) Describe the meaning of an adiabatic process. Using the first law of thermodynamics, derive the equation for an adiabatic process in the -plane for an ideal gas.
(d) Consider a simplified Otto cycle (an idealised petrol engine) involving an ideal gas and consisting of the following four reversible steps:
Adiabatic compression from volume to volume ;
: Heat injected at constant volume;
Adiabatic expansion from volume to volume ;
Heat extracted at constant volume.
Sketch the cycle in the -plane and in the -plane.
Derive an expression for the efficiency, , where is the work out, in terms of the compression ratio . How can the efficiency be maximized?
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Paper 2, Section II,
2021 comment(a) What systems are described by microcanonical, canonical and grand canonical ensembles? Under what conditions is the choice of ensemble irrelevant?
(b) In a simple model a meson consists of two quarks bound in a linear potential, , where is the relative displacement of the two quarks and is a positive constant. You are given that the classical (non-relativistic) Hamiltonian for the meson is
where is the total mass, is the reduced mass, is the total momentum, is the internal momentum, and is the centre of mass position.
(i) Show that the partition function for a single meson in thermal equilibrium at temperature in a three-dimensional volume can be written as , where
and
Evaluate and evaluate in the large-volume limit .
What is the average separation of the quarks within the meson at temperature ?
You may assume that for ]
(ii) Now consider an ideal gas of such mesons in a three-dimensional volume .
Calculate the total partition function of the gas.
What is the heat capacity
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Paper 3, Section II, C
2021 comment(a) A gas of non-interacting particles with spin degeneracy has the energymomentum relationship , for constants . Show that the density of states, , in a -dimensional volume with is given by
where is a constant that you should determine. [You may denote the surface area of a unit -dimensional sphere by .]
(b) Write down the Bose-Einstein distribution for the average number of identical bosons in a state with energy in terms of and the chemical potential . Explain why .
(c) Show that an ideal quantum Bose gas in a -dimensional volume , with , as above, has
where is the pressure and is a constant that you should determine.
(d) For such a Bose gas, write down an expression for the number of particles that do not occupy the ground state. Use this to determine the values of for which there exists a Bose-Einstein condensate at sufficiently low temperatures.
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Paper 4, Section II, 35C
2021 comment(a) Explain what is meant by a first-order phase transition and a second-order phase transition.
(b) Explain why the (Helmholtz) free energy is the appropriate thermodynamic potential to consider at fixed and .
(c) Consider a ferromagnet with free energy
where is the temperature, is the magnetization, and are constants.
Find the equilibrium value of at high and low temperatures. Hence, evaluate the equilibrium thermodynamic free energy as a function of and compute the entropy and heat capacity. Determine the jump in the heat capacity and identify the order of the phase transition.
(d) Now consider a ferromagnet with free energy
where are constants with , but .
Find the equilibrium value of at high and low temperatures. What is the order of the phase transition?
For determine the behaviour of the heat capacity at high and low temperatures.
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Paper 1, Section II, 30K
2021 comment(a) What does it mean to say that a stochastic process is a martingale with respect to a filtration ?
(b) Let be a martingale, and let for . Suppose takes values in the set almost surely for all . Show that is a simple symmetric random walk, i.e. that the sequence is with
(c) Let be a martingale and let the bounded process be previsible.
Let and
Show that is a martingale.
(d) Let be a simple symmetric random walk with , and let
where is a positive integer. Let
Show that is a simple symmetric random walk.
(e) Let be a simple symmetric random walk with , and let . Compute for a positive integer .
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Paper 2, Section II, 30K
2021 commentConsider a one-period market model with risky assets and one risk-free asset. Let denote the vector of prices of the risky assets at time and let be the interest rate.
(a) What does it mean to say a portfolio is an arbitrage for this market?
(b) An investor wishes to maximise their expected utility of time-1 wealth attainable by investing in the market with their time- 0 wealth . The investor's utility function is increasing and concave. Show that, if there exists an optimal solution to the investor's expected utility maximisation problem, then the market has no arbitrage. [Assume that is integrable for any attainable time-1 wealth .]
(c) Now introduce a contingent claim with time-1 bounded payout . How does the investor in part (b) calculate an indifference bid price for the claim? Assuming each such claim has a unique indifference price, show that the map is concave. [Assume that any relevant utility maximisation problem that you consider has an optimal solution.]
(d) Consider a contingent claim with time-1 bounded payout . Let be the set of initial no-arbitrage prices for the claim; that is, the set consists of all such that the market augmented with the contingent claim with time- 0 price has no arbitrage. Show that . [Assume that any relevant utility maximisation problem that you consider has an optimal solution. You may use results from lectures without proof, such as the fundamental theorem of asset pricing or the existence of marginal utility prices, as long as they are clearly stated.]
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Paper 3, Section II, 29K
2021 comment(a) Let be a martingale and a supermartingale. If , show that for any bounded stopping time . [If you use a general result about supermartingales, you must prove it.]
(b) Consider a market with one stock with time- price and constant interest rate . Explain why a self-financing investor's wealth process satisfies
where is the number of shares of the stock held during the th period.
(c) Given an initial wealth , an investor seeks to maximize , where is a given utility function. Suppose the stock price is such that , where is a sequence of independent copies of a random variable . Let be defined inductively by
with terminal condition for all .
Show that the process is a supermartingale for any trading strategy . Suppose that the trading strategy with corresponding wealth process are such that the process is a martingale. Show that is optimal.
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Paper 4, Section II, 29K
2021 comment(a) What does it mean to say that a stochastic process is a Brownian motion? Show that, if is a continuous Gaussian process such that and for all , then is a Brownian motion.
For the rest of the question, let be a Brownian motion.
(b) Let and for . Show that is a Brownian motion. [You may use without proof the Brownian strong law of large numbers: almost surely as .]
(c) Fix constants and . Show that
for any bounded function of the form
for some fixed and fixed , where is the space of continuous functions on . [If you use a general theorem from the lectures, you should prove it.]
(d) Fix constants and . Show that
for any bounded function . [In this part you may use the Cameron-Martin theorem without proof.]
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Paper 1, Section I, 2H
2021 commentWrite
and suppose that is a non-empty, closed, convex and bounded subset of with . By taking logarithms, or otherwise, show that there is a unique such that
for all .
Show that for all .
Identify the point in the case that has the property
and justify your answer.
Show that, given any , we can find a set , as above, with .
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Paper 2, Section I,
2021 commentLet be a non-empty bounded open set in with closure and boundary and let be a continuous function. Give a proof or a counterexample for each of the following assertions.
(i) If is twice differentiable on with for all , then there exists an with for all .
(ii) If is twice differentiable on with for all , then there exists an with for all .
(iii) If is four times differentiable on with
for all , then there exists an with for all .
(iv) If is twice differentiable on with for all , then there exists an with for all .
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Paper 3 , Section I,
2021 commentState Runge's theorem on the approximation of analytic functions by polynomials.
Let . Establish whether the following statements are true or false by giving a proof or a counterexample in each case.
(i) If is the uniform limit of a sequence of polynomials , then is a polynomial.
(ii) If is analytic, then there exists a sequence of polynomials such that for each integer and each we have .
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Paper 4, Section I, 2H
2021 comment(a) State Brouwer's fixed-point theorem in 2 dimensions.
(b) State an equivalent theorem on retraction and explain (without detailed calculations) why it is equivalent.
(c) Suppose that is a real matrix with strictly positive entries. By defining an appropriate function , where
show that has a strictly positive eigenvalue.
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Paper 2, Section II, H
2021 commentLet be a continuous function with for all but finitely many values of .
(a) Show that
defines an inner product on .
(b) Show that for each there exists a polynomial of degree exactly which is orthogonal, with respect to the inner product , to all polynomials of lower degree.
(c) Show that has simple zeros on .
(d) Show that for each there exist unique real numbers , such that whenever is a polynomial of degree at most ,
(e) Show that
as for all .
(f) If is real with and , show that
(g) If and , identify (giving brief reasons) and the . [Hint: A change of variable may be useful.]
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Paper 4, Section II, H
2021 commentLet be irrational with th continued fraction convergent
Show that
and deduce that
[You may quote the result that lies between and ]
We say that is a quadratic irrational if it is an irrational root of a quadratic equation with integer coefficients. Show that if is a quadratic irrational, we can find an such that
for all integers and with .
Using the hypotheses and notation of the first paragraph, show that if the sequence is unbounded, cannot be a quadratic irrational.
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Paper 1, Section II, 40A
2021 commentCompressible fluid of equilibrium density , pressure and sound speed is contained in the region between an inner rigid sphere of radius and an outer elastic sphere of equilibrium radius . The elastic sphere is made to oscillate radially in such a way that it exerts a spherically symmetric, perturbation pressure on the fluid at , where and the frequency is sufficiently small that
You may assume that the acoustic velocity potential satisfies the wave equation
(a) Derive an expression for .
(b) Hence show that the net radial component of the acoustic intensity (wave-energy flux) is zero when averaged appropriately in a way you should define. Interpret this result physically.
(c) Briefly discuss the possible behaviour of the system if the forcing frequency is allowed to increase to larger values.
For a spherically symmetric variable
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Paper 2, Section II, 40A
2021 commentA semi-infinite elastic medium with shear modulus and shear-wave speed lies in . Above it, there is a layer of a second elastic medium with shear modulus and shear-wave speed . The top boundary is stress-free. Consider a monochromatic SH-wave propagating in the -direction at speed with wavenumber .
(a) Derive the dispersion relation
for trapped modes with no disturbance as .
(b) Show graphically that there is always a zeroth mode, and show that the other modes have cut-off frequencies
where is a positive integer. Sketch a graph of frequency against for the mode showing the behaviour near cut-off and for large .
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Paper 3, Section II, 39A
2021 commentConsider a two-dimensional stratified fluid of sufficiently slowly varying background density that small-amplitude vertical-velocity perturbations can be assumed to satisfy the linear equation
and is a constant. The background density profile is such that is piecewise constant with for and with in a layer of uniform density .
A monochromatic internal wave of amplitude is incident on the intermediate layer from , and produces velocity perturbations of the form
where and .
(a) Show that the vertical variations have the form
where and are (in general) complex amplitudes and
In particular, you should justify the choice of signs for the coefficients involving .
(b) What are the appropriate boundary conditions to impose on at to determine the unknown amplitudes?
(c) Apply these boundary conditions to show that
where .
(d) Hence show that
where is the angle between the incident wavevector and the downward vertical.
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Paper 4, Section II, 39A
2021 commentA plane shock is moving with speed into a perfect gas. Ahead of the shock the gas is at rest with pressure and density , while behind the shock the velocity, pressure and density of the gas are and respectively.
(a) Write down the Rankine-Hugoniot relations across the shock, briefly explaining how they arise.
(b) Show that
where and is the ratio of the specific heats of the gas.
(c) Now consider a change of frame such that the shock is stationary and the gas has a component of velocity parallel to the shock on both sides. Deduce that a stationary shock inclined at a 45 degree angle to an incoming stream of Mach number deflects the flow by an angle given by
Note that
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Paper 3, Section II,
2021 commentConsider the normal linear model where is a known design matrix with is an unknown vector of parameters, and is a vector of normal errors with each component having variance . Suppose has full column rank.
(i) Write down the maximum likelihood estimators, and , for and respectively. [You need not derive these.]
(ii) Show that is independent of .
(iii) Find the distributions of and .
(iv) Consider the following test statistic for testing the null hypothesis against the alternative :
Let be the eigenvalues of . Show that under has the same distribution as
where and are independent random variables, independent of .
[Hint: You may use the fact that where has orthonormal columns, is an orthogonal matrix and is a diagonal matrix with
(v) Find when . [Hint: If with , then .]
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Paper 4, Section II,
2021 commentSuppose we wish to estimate the probability that a potentially biased coin lands heads up when tossed. After independent tosses, we observe heads.
(a) Write down the maximum likelihood estimator of .
(b) Find the mean squared error of as a function of . Compute .
(c) Suppose a uniform prior is placed on . Find the Bayes estimator of under squared error loss .
(d) Now find the Bayes estimator under the , where . Show that
where and depend on and .
(e) Determine the mean squared error of as defined by .
(f) For what range of values of do we have ?
[Hint: The mean of a Beta distribution is and its density at is , where is a normalising constant.]
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Paper 1, Section I, D
2021 commentLet be a bounded region of , with boundary . Let be a smooth function defined on , subject to the boundary condition that on and the normalization condition that
Let be the functional
Show that has a stationary value, subject to the stated boundary and normalization conditions, when satisfies a partial differential equation of the form
in , where is a constant.
Determine how is related to the stationary value of the functional . Hint: Consider .]
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Paper 3, Section I, D
2021 commentFind the function that gives a stationary value of the functional
subject to the boundary conditions and .
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Paper 2, Section II, D
2021 commentA particle of unit mass moves in a smooth one-dimensional potential . Its path is such that the action integral
has a stationary value, where and are constants, a dot denotes differentiation with respect to time
is the Lagrangian function and the initial and final positions and are fixed.
By considering for suitably restricted functions , derive the differential equation governing the motion of the particle and obtain an integral expression for the second variation .
If is a solution of the equation of motion and is also a solution of the equation of motion in the limit , show that satisfies the equation
If satisfies this equation and is non-vanishing for , show that
Consider the simple harmonic oscillator, for which
where is the oscillation period. Show that the solution of the equation of motion is a local minimum of the action integral, provided that the time difference is less than half an oscillation period.
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Paper 4 , Section II, 13D
2021 comment(a) Consider the functional
where , and is subject to the requirement that and are some fixed constants. Derive the equation satisfied by when for all variations that respect the boundary conditions.
(b) Consider the function
Verify that, if describes an arc of a circle, with centre on the -axis, then .
(c) Consider the function
Find such that subject to the requirement that and , with . Sketch the curve .